







Amy and Joey on a circular track 
20180813 

From Zoe: Amy and Joey are situated on a circular track 400 feet around.
Joey starts running at a rate of 10 feet per second.
Amy waits for a full minute, then starts running from the same point (in the same direction) at 12 feet per second.
How many seconds elapse before Amy passes Joey on the track? Answered by Penny Nom. 





A cyclist reduces his speed 
20180227 

From Katherine: Suppose the winner had reduced his average cycling rate by .1 mile per hour.
By how much would his time have changed?
(the cyclist's rate is 23.66 miles per hour and the answer to the question before this
one was that he cycled 2,292 miles in 96.87 hours) Answered by Penny Nom. 





What is the speed of the skateboarder? 
20180213 

From Zack: It takes a skateboarder 20 minutes to get to school, which is 6 km from home. What is the speed of the skateboarder? Answered by Penny Nom. 





Related rates 
20180211 

From angelo: hi admin please help me answer this question. thank you!
At a certain instant of time, the angle A of a triangle ABC is 60 degrees and increasing at the rate of 5degrees per second, the side AB is 10cm and increasing at the rate of 1cm per second, and side AC is 16cm and decreasing at the rate of 1/2 cm per second. Find the rate of change of side AB? Answered by Penny Nom. 





Heat equation 
20171123 

From Max: What does du\dt=a(triangle)^2u mean. Can it be solved for t. Answered by Penny Nom. 





How many miles did he drive in one hour? 
20171121 

From Ava: Graham drove 42 1/3 Miles in 1 1/3 hours. How many miles did he drive in one hour? Answered by Penny Nom. 





The distance between the origin and a moving point on a graph 
20171016 

From Paulina: Find the rate of change of the distance between the origin and a moving point on the graph of y=x^2 +1 if dx/dt=2 centimeters per second Answered by Penny Nom. 





Walking up and down a hill 
20170919 

From Justina: At 2:20 p.m. Jack is at the top of the hill and starts walking down at the exact same time that Jill, who is at the bottom of the hill, starts walking up. If they maintain the uphill speed of 2mph and downhill speed of 4mph and the distance from the bottom to the top of the hill is 1.5 mi, at what time will Jack and Jill meet? Answered by Penny Nom. 





Up a hill and down again 
20170917 

From justina: Jack and Jill travel up a hill at a speed of 2mi/h.they travel back down the hill at a speed of 4mi/h.what is their average speed for the entire trip? express your answer as a mixed number Answered by Penny Nom. 





A car on a rough road and a concrete road 
20170827 

From Ken: A car is driven on a rough road at a speed of 25km and then on a concrete road with a speed of 70kph. The car has 5.5 hours to travel 225km. How long did the car travel on concrete road? Answered by Penny Nom. 





A police officer spots a speeding car 
20170720 

From Josh: Marie, a police officer, spots a speeding car and starts chasing it. The speeding
car travels at a speed of 130 feet per second. Marie's car reaches a constant speed
of 145 feet per second 1725 feet from the start of the chase. During that time, the
speeding car has traveled 3150 feet. Write a system of equations to represent this
situation. Answered by Penny Nom. 





How fast am I travelling? 
20170624 

From corey: if i travel 2 miles in 10 seconds, how fast am i travelling please Answered by Penny Nom. 





Initial velocity 
20170523 

From Annelle: a body is thrown vertically downward from a height above the ground. determine the initial velocity and the height of the starting point if after 4 secs it reaches the ground with velocity of 68 m/s. Answered by Penny Nom. 





The length of a train 
20170424 

From Stefano: If a freight train is coming towards you and you were traveling at 85 miles an hour and it takes 35 seconds for the train to pass you how long is the train
Figure 50mph for train Answered by Penny Nom. 





How much money do I make per sq ft? 
20170328 

From jackie: If I make $12.00 per hour and I cover 750 sq ft in one hour, how much money do I make per sq ft? Answered by Penny Nom. 





Filling a pool with two hoses 
20170220 

From Charlene: A large hose will fill a pool in 40 minutes while it takes a smaller hose 60 minutes to fill the same pool. The same pool takes 80 minutes to drain if the drain is open. Suppose one day both the hoses are turned on, but by accident the drain was left open. How long would it take to fill the pool that day? Answered by Penny Nom. 





The average rate of change of cot(t) 
20170118 

From Brianna: Hello!
It's been a while since I've taken a math course, and I'm stuck on a problem in my calculus course.
The question is this:
Find the average rate of change of the function over the given interval.
h(t)=cot(t) a) [5pi/4, 7pi/4] Answered by Penny Nom. 





Water leaking from a trough 
20161228 

From Kathryn: A trough is 6 m long, and has uniform crosssection of an equilateral triangle with sides 1 m.
Water leaks from the bottom of the trough, at a constant rate of 0.1 m3/min.
Find the rate at which the water level is falling when the water is 0.2m deep. Answered by Penny Nom. 





Travelling upstream and downstream 
20161017 

From Kathleen: It took a boat 2 hours to reach town A going upstream. The way back was 1h20 min. What is the speed of the boat in still water if the speed of the stream is 3 mph? Answered by Penny Nom. 





A car chase 
20160925 

From Ms.: Car 1 is traveling down the road at 50 mph and passes Car 2 is
sitting on the side of the road. Car 2 needs to catch Car 1 one half mile
down the road. How fast must Car 2 drive to catch Car 1 if Car 2
waits 10 seconds before pursuing Car 1?
Yes, the assumption is that the Car 2 can accelerate from a standing position and over take Car 1 rate. What is the answer or formula for this type of question?
Respectfully
Befuddled Mom? Answered by Penny Nom. 





How fast and how far must Car 2 travel before catching Car 1? 
20160925 

From Ms.: Car 1 is traveling down the road at 50 mph and passes Car 2 is
sitting on the side of the road. Car 2 needs to catch Car 1 one half mile
down the road. How fast must Car 2 drive to catch Car 1 if Car 2
waits 10 seconds before pursuing Car 1?
Yes, the assumption is that the Car 2 can accelerate from a standing position and over take Car 1 rate. What is the answer or formula for this type of question?
Respectfully
Befuddled Mom? Answered by Penny Nom. 





Integrate (x^2  4x + 4) ^4/3 
20160915 

From Ifah: Hi i have questions please answer
Integral 2 sampai 3 dari (x²  4x + 4) ^4/3 dx Answered by Penny Nom. 





John and Max walk to meet each other 
20160529 

From abhay: The speeds of John and Max are 30 km/h and 40 km/h.Initially Max is at a place L and John is at a place M. The distance between L and M is 650 km. John started his journey 3 hours earlier than Max to meet each other. If they meet each other at a place P somewhere between L and M, then the distance between P and M is? Answered by Penny Nom. 





Two tax rates 
20160421 

From Vishal: Total amount paid annually $444 for 2014 which is inclusive for tax.
tax rate for first six month is 7% then it is changed to 15% for second six months.
calculate the total basic amount paid before tax.
the basic amount on which tax paid in first and second six month is same. Answered by Penny Nom. 





Biking and jogging 
20160317 

From alondra: Ricky went on a bike ride. After 6 miles he got a flat tire and had to jog home. He jogs 3mph slower than he bikes, so the jog took 1 hour longer than the bike ride. At what rate did he travel each way? Answered by Penny Nom. 





Travelling with and against he current 
20160209 

From Lloyd: A boat travelled 210 miles downstream and back. The trip downstream took 10hrs, the trip back took 70hrs
1. what is the speed of the current?
2. what is the speed of the boat in still water? Answered by Penny Nom. 





A two car competition 
20160202 

From Christian: Question: Two cars are racing to a fixed point and back. One car travels 70 MPH both ways
While the other travels 60 MPH there and 80 MPH back, which car won the race?
It is not a tie. Answered by Penny Nom. 





Riding a mountain bike up and down a mountain 
20151202 

From sam: Michael, can climb a mountain at 2.5km/hr on his mountain bike. he descends the mountain at 5.5km/hr. he completes the whole circuit in 7 hours. how long is the trail to the top of the mountain? Answered by Penny Nom. 





Two lorries approaching an intersection 
20150515 

From Nuraini: Two straight roads intersect at the right angles. Lorry A, moving on one of the roads,
Approaches the intersection at 50mi/h and lorry B, moving on the other roads, approaches the intersection at 20mi/h.
At what rate is the distance between the lorry changing when A is 0.4 mile from the intersection and B is 0.3 mile from the intersection? Answered by Penny Nom. 





Two cars approaching an intersection 
20150416 

From Engabu: Car A is traveling west at 50km/h & car B is traveling north at 60km/h. both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 3km & car B is 4km from the intersection? Answered by Penny Nom. 





A train trip 
20150323 

From Ngcebo: A train travels a distance of 720 km at a speed of x km/h. on the return journey, it travels at an average speed of y km/h and completes the whole journey in an hour less. if the speed during the return journey was 10 km/h more, what were the speeds in m/s Answered by Penny Nom. 





Istanbul to Lagos 
20150303 

From Victor: A plane flies from Istanbul to Lagos at 800km/h and then returns to Istanbul at 900km/h. The return trip takes 1 hour less than the flight to Lagos. What is the distance between Istanbul and Lagos? Answered by Penny Nom. 





Distance, time and rate for three people 
20150215 

From Mubashir: Steven left Town A and walked towards Town B at a speed of 100m/min. At the same time, Jason and Melvin started from Town B and walked towards Town A at a speed of 80m/min and 75m/min respectively. If Steven met Melvin six minutes after passing Jason, find the distance between Town A and Town B. Answered by Penny Nom. 





Two cars pass each other going in opposite directions 
20150203 

From Caroline: Two cars pass each other going i opposite directions. One car is going 100 km/h and the other is going 80 km/h. How far apart will the two cars be after driving 45 minutes at these rates of speed? Answered by Penny Nom. 





A motorbike race 
20150125 

From Pramegh: In a motorbike race of 75 km, the winner completes the race with a lead of 300 meters and 8 seconds before the second rider. Considering the riders ride with an average speed, find the difference between their speeds with which they completed the race.
(Calculate up to 2 decimal places) Answered by Penny Nom. 





Two cars 
20150118 

From taylor: two cars leave the park at the same time, one traveling north at a speed 10 mph faster than the other which is traveling south. after 3 hours the cars are 420 miles apart. find the speed of both cars Answered by Penny Nom. 





Pumping water from a borehole 
20150118 

From patience: An empty tank for storing water from a borehole has a volume of 480 cubic metres. If it is filled by a pump that pumps water at a rate of 16 litres per second, how many hours will it take the pump to fill this tank please show all calculations Answered by Penny Nom. 





Filling a tub 
20150102 

From Mr: The cold tap can fill a bath in 5minutes,the hot tap in 15,and the waste pipe empty the full bath in 10 minutes. All three are fully open for 2 minutes,after which the wastepipe is closed. How much longer will it take to fill the bath? Answered by Penny Nom. 





Rates, percentages and units 
20141230 

From Kenneth: Hello:
If percentages have no units, why are some percentages called rates, as in interest rate, or
perhaps a tax rate of 7% as an example? A rate has units of different quantities.
I thank you for your reply. Answered by Robert Dawson. 





196 kilometer in2 hours 
20141229 

From Ekanem: QQ A motorist had planned to reach a place 196km in two hrs exactly. For the first 15km he was able to average only 40km/h. What average speed must he keep up for the rest of the journey if he is to arrive on time. Answered by Penny Nom. 





A man a ship and a wharf 
20140926 

From Ralph: A man om wharf 12 meters above the level of water is pulling a rope tide to a boat at the rate of 2 meters per minute. How fast is the boat approaching the wharf when there are 20 meters of the rope our. (with solution) Answered by Penny Nom. 





What is the speed of the buses? 
20140518 

From musliu: Two buses leave a bus station and travel in opposite direction from that same starting point.if the speed of one is twice the speed of the other and they are 240km apart at the end of 1h, what is the speed of the buses? Answered by Penny Nom. 





Travelling upstream and downstream 
20140512 

From Mohanad: Howie Sorkin can travel 8 miles upstream in the same time it takes him to go 12 miles downstream. His boat goes 15 mph in still water. What is the rate of the current? Answered by Penny Nom. 





2 cars driving towards each other 
20140427 

From Joakim: 2 cars driving towards eachother , the distance between them is 162km. they meet after 2 hours. car A is driving with 4/5 of car B´s speed. what is the speed of each car... Answered by Penny Nom. 





What time was it when joe's brother passed him? 
20140425 

From Nathan: joe left home in his bike at 10:00 am, traveling 21 km/h. At noon, his brother set out after him on his motorcycle, following the same route. if the motorcycle traveled at 63 km/h, what time was it when joe's brother passed him? Answered by Penny Nom. 





How fast did Jessica grow? 
20140410 

From jasmine: when she was 14 jessica was 5 feet, 4inches tall . at 18 she is now 6 feet, 2inches tall .how fast did jessica grow during this time Answered by Robert Dawson. 





A speed word problem 
20140224 

From kiss: Hank bicycles 5 km/h slower than Kelly. In the time that it takes Hank to bicycle 42 km, Kelly can bicycle 57 km. How fast does each bicyclist travel? Answered by Penny Nom. 





Water in a conical funnel 
20140211 

From Marcus: Water is running out of a conical funnel at the rate of 1 inch^3/sec. If the radius of the base of the funnel is 4 in. and the altitude is 8 in., find the rate at which the water level is dropping when it is 2 in. from the top. Answered by Penny Nom. 





Two bicyclists traveled in opposite directions 
20140207 

From Susan: Two bicyclists traveled in opposite directions, one at a rate of 8 km/hour and the other at a rate of 7 km per hour. In how many hours were they 45 km apart? Answered by Penny Nom. 





Walking up and down a hill 
20140206 

From flo: michelle walks 4 km to the top of a hill at 3 km/h then immediately walks back down at an average speed of 5 km/h what is her average speed for the 8km walk Answered by Penny Nom. 





Related rates 
20140130 

From Veronica: A container is the shape of an inverted right circular cone has a radius of 1.00 inches at the top and a height of 5.00 inches. At the instant when the water in the container is 1.00 inches deep, the surface level is falling at the rate of 2.00 inches/second. Find the rate at which the water is being drained. Answered by Penny Nom. 





A man and a kite 
20140129 

From Veronica: A man flies a kite at a height of 120 meters. The wind carries the kite horizontally away from him at a rate of 8 meters/second. How fast is the distance between the man and the kite changing when the kite is 130 meters away from him? Answered by Penny Nom. 





Two tuna boats 
20140109 

From amanda: Two tuna boats start from the same port at the same time, but they head in opposite directions. The faster boat travels 10 knots per hour faster than the slower boat. At the end of 8 hours, they were 272 nautical miles had each boat traveled by the end of the 8 hour period? Answered by Penny Nom. 





One train overtaking another 
20131111 

From kiran: two trains of equal length are running in parallel tracks in the same direction with the speed of 60km/hr and 90km/hr respectively the latter completely crosses the former in 30 seconds the length of each train is in mtr Answered by Penny Nom. 





Water flowing out of a tank 
20131103 

From Carolyn: The flow of water out of a hole in a tank is known to be proportional to the square root of the height of water above the hole.
That is,
dV/dt (proportional to) sq root (h)
The tank has a constant crosssectional area A, show that the height of water in the tank is given by
h = ((kt+C)/2)^2
If the tank is 9 metres high, and it takes 5 hours for it to drain from full to half full,
how much longer will we have to wait until it is completely empty? Answered by Penny Nom. 





What is the original speed? 
20130810 

From MJ: If a truck increases its average speed by 10 km/h it can cover 60 km at the original speed in the same time that it can cover 80 km at the faster speed. What is the original speed? Answered by Penny Nom. 





Diego and his dog 
20130709 

From William: john went hunting and walked at a speed of 4 km/h. one hour later, Diego and his dog went to meet john with a speed of 6 km/h and the dog was running between john and diego until they meet up with a speed of 18 km/h. After Diego meets john, how much distance did the dog run? Answered by Penny Nom. 





An isosceles tiangle 
20130616 

From Izzy: what's the height of an isosceles triangle which has a base of 50 m, and both of the other sides are 25 m? Answered by Penny Nom. 





Related rates 
20130217 

From Ishaak: A hemispherical bowl is filled with water at a uniform rate. When the height of water is h cm the volume is π(rh^21/3 h^3 )cm^3, where r s the radius. Find the rate at which the water level is rising when it is half way to the top, given that r = 6 and the bowl fills in 1 minute. Answered by Penny Nom. 





An electron in a TV tube 
20130215 

From anu: an electron in a TV tube is beamed horizontally at a speed of (50^6) m/sec. towards the face of a tube 40 cm away
about how far will the electron drop before it hits? no information has been provided of initial height from where it is beamed. Answered by Robert Dawson. 





A journey of two parts 
20130206 

From Natalie: Mr Lim drove at 70 km per hour for 1 hour and 30 minutes. He then drove another 55 km. if his average speed for the whole journey was 80km per hour, what was his speed for the 55 km journey? Answered by Penny Nom. 





How fast are we approaching each other? 
20130111 

From Heather: If I am skating at 7 mph, and a car drives towards me at 15 mph, how fast are we approaching each other? Answered by Penny Nom. 





How fast is the distance between the aircraft and the car increasing? 
20121024 

From Steven: At a certain instant an aircraft flying due east at 240 miles per hour passes directly over a car traveling due southeast at 60 miles per hour on a straight, level road. If the aircraft is flying at an altitude of .5mile, how fast is the distance between the aircraft and the car increasing 36 seconds after the aircraft passes directly over the car? Answered by Penny Nom. 





Two swimmers 
20121022 

From Emily: Two swimmers start from opposite sides of the pool. Each swimmer swims the length of the pool and back at a constant rate. They pass each other for the first time 40 feet from one side of the pool and for the second time 45 feet from the other side of the pool. What is the length of the pool? Answered by Penny Nom. 





A problem involving simple interest 
20120927 

From Tash: The amount of money in a single year of an investment after P dollars were initially invested is A = P + Prt, where r is the rate of simple interest. What expression describes P? How much money was initially invested if the account has $1000 one year after the initial investment and the interest rate was 5%? Answered by Penny Nom. 





A cyclist a jogger and a fly 
20120604 

From Emd: A cyclist & a jogger are 20 miles apart.
Cyclist goes 20 mph & jogger 7mph towards each other.
A fly starts on nose of cyclist & flies at 20 mph from cyclist to jogger &
back & forth until they meet.
How far has the fly travelled? Answered by Penny Nom. 





Romeo and Juliet live 40 miles apart 
20120518 

From Abigaíl: Romeo and Juliet live 40 miles apart. Romeo jogs 8mph, while Juliet “fast walks” at 4mph. They leave their houses at the same time and journey toward each other.
a) How many hours after they start will they meet?
b) Approximately how many miles from Romeo’s house is their meeting point? Answered by Penny Nom. 





Math, time and rate 
20120518 

From Abigaíl: Romeo and Juliet live 40 miles apart. Romeo jogs 8mph, while Juliet “fast walks” at 4mph. They leave their houses at the same time and journey toward each other.
a) How many hours after they start will they meet?
b) Approximately how many miles from Romeo’s house is their meeting point? Answered by Penny Nom. 





Points per quarter 
20120514 

From debbie: A basketball team scores 216 points in 8 quarters . Find the teams point per quarters rate Answered by Penny Nom. 





Cycling and running 
20120507 

From liz: a biathlete travels 20 miles in 2.25 hours. She cycles part of the way at 12 mph and runs the rest at 5 mph. How far did she run? Answered by Penny Nom. 





Two cars approach a rightangled intersection 
20120410 

From Michael: Two cars approach a rightangled intersection, one traveling south a 40km/h and the other west at 70km/h.
When the faster car is 4km from the intersection and the other case if 3km from the intersection,
how fast is the distance between the car cars changing? Answered by Penny Nom. 





The spread of a rumor 
20120409 

From Roohi: The function f(t) = a/(1+3e^(bt)) has also been used to model the spread of a rumor. Suppose that a= 70 and b=3 0.2. Compute f(2), the percentage of the population that has heard the rumor after 2 hours. Compute f'(2) and describe what it represents. Compute lim t approaches infinity and describe what it represents. Answered by Penny Nom. 





Water is flowing from tank A to tank B 
20120406 

From Noel: At First, Tank A was completely filled with water. The tap from Tank A was then turned on and water flowed out at a constant rate. The water which flowed out was collected into Tank B. At the end of 6 minutes, Tank A was 2/3 full. After a further 8 minutes, Tank A had 2.4 liters of water left while Tank B was completely filled with water. Find the capacity of Tank B. Answered by Penny Nom. 





The period T of a pendulum 
20120327 

From Ashley: The period T of a pendulum is given in terms of its length, l, by T=2pi sqrt(l/g) where g is the acceleration due to gravity(a constant)
a. find dT/dl
b. what is the sign of dT/dl
c. what does the sign of dT/dl tell you about the period of the pendulums? Answered by Penny Nom. 





How long will it take lee to catch up with jodie? 
20120321 

From kirsten: for example, if jodie is on the bus travelling at 10 m.p.h and lee is walking at 4 m.p.h how long will it take lee to catch up with jodie?? what is this mathematical name for this Answered by Harley Weston. 





Margie threw a ball 
20120216 

From mary: at 9:45 Margie threw a ball upwards while standing on a platform 35ft above the ground. The height after t seconds follows the equation:
h(t)= 0.6t^2 +72t+35
a) what will be the maximum height of the ball?
b)how long will it take the ball reach its maximum height?? Answered by Harley Weston. 





Jack and Kate set off in their cars ... 
20120213 

From akiri: Jack and Kate set off in their cars from the same point to travel the same journey. Jack has a start of eight minutes before Katie sets off. If Jack travels at 50mph and Katie travels at 75mph, how many miles will be travelled when the two cars are level? Answered by Penny Nom. 





There is a police officer in pursuit of suspect 
20120201 

From Jay: There is a police officer in pursuit of suspect. They suspect is driving on a straight road traveling at 79 mph, the officer is 233 yds away, traveling at 92 mph, the suspects car 5 ft long and the officers car is 5.3 ft long, how long will it take to have the front of the officers car, equal to the front of the suspects car down to the tenth of a second. Answered by Harley Weston. 





Two cars in a race 
20120130 

From Christine: In a car race, the speed of one car is 120kph and the other car is 105 kph. If the faster car finished the race 20 mins before the slower one, what was the distance of the race? Thanks for the answer and the previous answer to my question. Answered by Penny Nom. 





Brian on his bike 
20120116 

From tiffany: Brian rides his bike 0.29 miles per minute. If it takes him 22 minutes to ride his bike to his friend's house, how far away does his friend live? Answered by Penny Nom. 





Running 5 km 
20120112 

From Fayeann: What amount of time will it take a person running 7 m/s to travel a distance of 5 km? Answered by Penny Nom. 





Water is flowing into a cup 
20111219 

From Tim: A cup has a radius of 2" at the bottom and 6" on the top. It is 10" high. 4 Minutes ago, water started pouring at 10 cubic " per minute. How fast was the water level rising 4 minutes ago? How fast is the water level rising now? What will the rate be when the glass is full? Answered by Penny Nom. 





Simple interest 
20111211 

From sandeep: Use the formula I = Prt to solve.
Damon deposits $500 into a savings account that pays simple interest
at a rate of 0.65% per year. How long will it take Damon to earn $130
in interest? Answered by Penny Nom. 





Two trains passing each other 
20111207 

From Tamkeen: two trains 245m and 315m long are travelling toward each other at 90km/h and 54km/h respectively on parallel lines . how long do the train take to pass one another train the time they meet each other? Answered by Penny Nom. 





Uniform acceleration 
20111206 

From Android: At a certain instant, two cars A and B are 2000 ft apart. At this instant, car A is traveling at 15 miles/hr. and accelerating at 3 ft/s^2 while B is moving towards A at 30 miles/hr. and an acceleration of 2 ft/s^2.
Find the ff:
a.) The possible time of their collusion in minutes.
b.) The distance in ft. each has traveled before collusion. Answered by Penny Nom. 





A cube of ice is melting 
20111205 

From Emily: a cube of ice (i.e.) each side is of the same length) is melting at a rate such that the length of each side is decreasing at a rate of 5cm per hour. how fast is the volume of the cube decreasing (in cubic cm per hour) at the instant the length of each side is 25cm? Answered by Penny Nom. 





Two boats 
20111130 

From Shayan: A boat is traveling 20 miles per hour when it passes a lighthouse.
another boat is traveling at 15 miles per hour when it passes a point.
the lighthouse and the point are 175 miles apart.
when will the boats pass each other? Answered by Penny Nom. 





Four carpenters can build eight houses in 10 days. 
20111123 

From Kenneth: Four carpenters can build eight houses in 10 days.
Two carpenters can build how many houses in 15 days? Answered by Penny Nom. 





Water pouring into a conical tank 
20111121 

From Patience: Hi my name is patience and I'm having a problem with this question.
Water pours into a conical tank of semi vertical angle 30 degrees at the rate of 4 cm^3/s, where h is the depth of the water at time t. At what rate is the water rising in the tank when h = 10 cm?
Thank you Answered by Penny Nom. 





Chuong and Hassan drive to work 
20111004 

From Dilshad: Chuong and Hassan both drive 40 km from home to work each day. One day Chuong said to Hasssan, "If you drive home at your usual speed, I will average 40 kmph faster than you and arrive home in 20 minutes less time." Find Hassan's speed. Answered by Penny Nom. 





A reservoir has the shape of an inverted cone 
20111003 

From Roger: a reservoir has the shape of an inverted cone whose cross section is an equilateral triangle. if water is being pumped out of the reservoir at a rate of 2m^3/sec, at what rate is the depth of the water changing when the depth is 40 meters? Answered by Penny Nom. 





A hemispherical bowl with a lead ball inside 
20110927 

From Jean: "(a) Water is being poured into a hemispherical bowl of radius 3 inch
at the rate of 1 inch^3/s. How fast is the water level rising when the
water is 1 inch deep ?
(b) In (a), suppose that the bowl contains a lead ball 2 inch in
diameter, and find how fast the water level is rising when the ball is
half submerged." Answered by Penny Nom. 





Driving to meet Mary 
20110824 

From Denise: mary leaves St Paul traveling 2000 miles @ 55 MPH; John leaves LA traveling 2000 miles @ 45 MPH how many miles will John had traveled to meet mary? Answered by Penny Nom. 





The height of a fluid in a horizontal tank 
20110724 

From jason: Same set up as many others, cylindrical tank on its side, but I am interested in defining the change in volume and/or fluid level as a function of time at a constant volumetric outflow. I plan on hooking a pump to the tank so "gpms' will be constant. I have a couple different sized tanks and pumps so I want a general equation. Thanks for your help. Answered by Harley Weston. 





Achilles and a turtle 
20110701 

From Jean: Achilles and a turtle are having a race. The turtle starts 45m ahead of Achilles and Achilles is twice as fast as the turtle. If turtle runs at 1m/s,how far would the turtle have run before he is outrun by Achilles? Answered by Penny Nom. 





Two trains 
20110607 

From Lynn: If the 1st train goes 60 mph and the 2nd train leaves 10 minutes later =
at a speed of 70 mph, how long does it take to catch up with the 1st =
train. Answered by Penny Nom. 





How many meters long should the race be? 
20110526 

From Lee: Nathan can walk at a rate of two meters per second while David can
easily go threeandahalf meters per second. David offers Nathan
a 45 meter head start. How many meters long should the race be in
order for Nathan to win by a nose Answered by Penny Nom. 





Find the rate at which the searchlight rotates 
20110417 

From Meredith: A searchlight is position 10 meters from a sidewalk. A person is walking along the sidewalk at a constant speed of 2 meters per second. The searchlight rotates so that it shines on the person. Find the rate at which the searchlight rotates when the person is 25 meters from the searchlight. Answered by Penny Nom. 





A conical container and a spherical balloon 
20110406 

From Steven: Water is running out of a conical container 12 feet in diameter and 8 feet deep (vertex down) and filling a spherical balloon.
At the instant the depth of the water in the cone is 4 feet, the radius of the sphere is approximately 4 feet.
The rate of change of the depth of the water in the cone at the instant is approximately ______________ times the rate of change of the radius of the balloon. Answered by Penny Nom. 





Two ships 
20110405 

From Gevork: Ship A is sailing due south at 16 mph. At the same time, a second ship B, 32 miles south of A, is sailing due east at 12 mph.
(a) at what rate are they approaching or separating at the end of one hour?
(b) At what rate are they approaching or separating at the end of two hours?
(c) When do they cease to approach each other and how far apart are they at that instant. Answered by Penny Nom. 





The rate of change of the area of a parallelogram 
20110405 

From Gevork: Let a parallelogram have sides of 8 and 12 and let vertex angle A be decreasing at a rate of 2degrees per minute. Find the rate of change of the area of the parallelogram when angle A equals 30 degrees. Answered by Penny Nom. 





A stone is dropped into a lake 
20110324 

From AnneMarie: A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 25 cm/s. Find the rate at which the area within the circle is increasing after 4s. Answered by Penny Nom. 





A camera's line of sight 
20110226 

From MJ: A rocket that is rising vertically is being tracked by a ground level camera located 3 mi from the point of blast off when the rocket is 2 mi high its speed is 400mph At what rate is the (acute) angle between the horizontal and the camera's line of sight changing Answered by Penny Nom. 





At what rate is the grain pouring from the chute? 
20110226 

From MJ: Suppose that grain pouring from a chute forms a conical heap in such a way that the height is always 2/3 the radius of the base. At the moment when the conical heap is 3 m high, its height is rising at the rate of 1/2 m/min. At what rate (in m^3/min) is the grain pouring from the chute? Answered by Penny Nom. 





A player runs from second base to third base 
20110130 

From Marie: A baseball diamond is a square with side 90 feet in length. A player runs from second base to third base at a rate of 18 ft/sec. At what rate is the area of the trapezoidal region, formed by line segments A, B, C, and D changing when D is 22.5
Distance A is the players distance from first base when running from 2nd to third. Distance D is his distance from 3rd base. Distance C is the distance from 3rd to 3rd to Home. Distance B is the distance from Home to First.
I have found dA/dt in a previous problem. Answered by Penny Nom. 





How long will you and your friend be able to talk? 
20101123 

From beket: you and your friend leave your home town at the same time your friend travels by train 45 mph directly east while you travel by train 50 mph going directly south you are talking on cell phone but you plan charges extra if the phones are 100 miles apart. How long will you and your friend be able to talk before the charges increase? Answered by Penny Nom. 





The rate of change of (8e^3x)+(27 e^3x) 
20101123 

From Aleo: I am unable to solve this problem:
Find the rate of change of (8e^3x)+(27 e^3x), with respect to x when x= 0.5 Answered by Penny Nom. 





John and Mary are 200 miles apart 
20101020 

From Sarah: two people, john and mary, are 200 miles apart and leave at the same time to meet.
They are driving toward each other at constant rate on the same straight road.
John is traveling at 45 mph and Mary is traveling at 35 mph.
When they meet, how many miles will they be from john's house? Answered by Penny Nom. 





A commuter airline between two cities 
20101019 

From Adori: An airline runs a commuter between two cities that are 720 miles apart. If the average speed of the planes is increased by 40 miles per hour, the travel time is decreased An airline runs a commuter between two cities that are 720 miles apart. If the average speed of the planes is increased by 40 miles per hour, the travel time is decreased by 12 minutes. What airspeed is required to obtain this decrease in travel time? Answered by Penny Nom. 





Electronic cars around a circular track 
20101011 

From Taylor: Cory and Melissa are racing electronic cars around a circular track. They begin at the same time going in the same direction. Cory's car completes a revolution in 35 secs, while Melissa's car completes a revolution in 30 secs. How long will it take them before both cars reach the starting point again simultaneously? Answered by Stephen La Rocque and Penny Nom. 





Two cars 
20100930 

From Amanda: Two cars, 142 miles apart, start moving towards each other at the same
time. One is moving 3 times as fast as the other. If they meet 1.7 hrs
later, find the average speed of the slower car in miles per hour? Answered by Penny Nom. 





The distance between the origin and a moving point 
20100924 

From Norma: I am having problems with this question
find the rate of change of the distance between the origin and a moving point on the graph of the function below if dx/dt=5 cm/sec
y=x^2+2 Answered by Penny Nom. 





40 mph for 6 minutes 
20100920 

From Andy: If i was traveling at 40 mph for 6 minutes, what is my distance in miles? Answered by Penny Nom. 





Three cars 
20100830 

From Anil: 3 cars are moving at speed of 4 kmph,5.5 kmph and 8 kmph in a circular track.The circular track has
a distance of 11 km. What is the time taken for all the 3 cars to meet at the starting point ? Answered by Stephen La Rocque. 





Running 100 km 
20100812 

From thomas: Another athlete runs 8 laps every day on the same 400m running track. How many days will it take him to run a total of 100 km? Answered by Robert Dawson. 





How fast is the second car taveling? 
20100726 

From Jeff: A car is traveling 75 mph. It takes 12 miles to overtake a second car that was .5 mile ahead. How fast is the second car taveling? Answered by Stephen La Rocque. 





How much longer will it take to reach Toronto 
20100722 

From Rachelle: We are taking a trip from Ottawa, ON to Toronto, ON. The total trip is 150 kms and will take us 4.5 hours. We have already driven 108 kms and there is still 42 kms left to go. If we are travelling 110 kms/hr, how much longer will it take to reach Toronto? Answered by Penny Nom. 





Pursuing Car 1 
20100713 

From Mark: Car 1 is traveling down the road at 50mph and passes Car 2 is
sitting on the side of the road. Car 2 needs to catch Car 1 one half mile
down the road. How fast must Car 2 drive to catch Car 1 if Car 2
waits 10 seconds before pursuing Car 1? Answered by Robert Dawson. 





Half a mile at 50 mph 
20100713 

From Mark: How long will it take a car to travel 1/2 mile at 50 mph? Answered by Robert Dawson. 





Fillig the pool 
20100629 

From saurabh: At certain swimming pool certain pipes can fill it in 2 hours, another can fill it in 5 hours,and third pipe can empty the pool in 6 hours. with all three pipes turned on at the same time and starting with an empty pool , how long will it take to fill the pool Answered by Robert Dawson. 





Going to the bike shop 
20100607 

From Omi: Marie rode her bicycle from her home to the bicycle shop in town and then walked back home. If she averaged 6 miles per hour riding and 3 miles per hour walking, how far is it from her home to the bicycle shop if her total travel time was 1 hour? Answered by Penny Nom. 





What is the distance from A to C via B? 
20100524 

From vivianne: The distance from A to B is d km and that from B to C is x km. if a bus maintains an average speed of 50km/hr between A and B and 60km/hr between B and C, it takes 3 hours to travel from A to C. If it maintains 60km/hr between A and B and 50km/hr between B and C, the journey takes 8 minutes less. What is the distance from A to C via B? Answered by Penny Nom. 





A circular oil slick of uniform thickness 
20100522 

From Susan: Hi,
I have this problem on a homework assignment and just can't seem to figure it out:
A circular oil slick of uniform thickness is caused by a spill of 1 m^3 of oil. The thickness of the oil is decreasing at the rate of .001m/h. At what rate is the radius of the slick increasing when the radius is 8. Answered by Penny Nom. 





Working together 
20100504 

From Felicia: Sarah takes 3 hours longer to paint a floor than it takes Kate. when they work together it takes 2 hours. How long would each take to do the job alone? Answered by Penny Nom. 





40 miles at 50mph 
20100502 

From gill: if a car travelled 40miles at 50mph, how long would it take? Answered by Penny Nom. 





When are the cars exactly 40 miles apart? 
20100501 

From Maryanne: A car traveling east at 45 mph passes a certain intersection at 3pm. Another car traveling north at 60mph passes the same intersection 25 minutes later. To the nearest minute, figure out when the cars are exactly 40 miles apart. Answered by Penny Nom. 





The rate of change of y with respect to x 
20100429 

From Tom: I just had a quick calc question about wording that wasn't ever
addressed in class. When the book says "the rate of change of y with
respect to x", should it be considered how fast y is changing in
comparison to x?
I ask because the textbook says that "y is changing 3 times faster than x,
so the rate of change of y with respect to x is 3." I'm use to rate being
like velocity, as in units of distance per units of time. All we're told
in class is that it's the slope of the tangent line, I was hoping you
could clarify for me what exactly is meant by the wording of a "rate of
change of something with respect to something else". More specifically, what
"rate" and "with respect to" mean within this context?
Thanks for your time Answered by Harley Weston. 





Integrate the ((4th root of x^3)+1) dx 
20100412 

From Bridget: integrate the ((4th root of x^3)+1) dx Answered by Tyler Wood. 





Related rates and a rectangular sponge 
20100406 

From Heather: A rectangular sponge is increasing its length at 4cm/min, decreasing its width at 2cm/min, and increasing its height at 3cm/min. When its length, width and height are 40, 30, and 20 respectively, find the rate of change of volume and surface area. Answered by Penny Nom. 





Sand falling off a conveyer 
20100402 

From Katherine: sand is falling off a conveyer onto a pile at the rate of 1.5 cubic feet per minute. The diameter of the base is approximately twice the altitude. At what rate is the height of the pile changing when it is 10 feet high? Answered by Penny Nom. 





Sand in an hourglass 
20100320 

From Luke:
Answered by Harley Weston. 





At what time would Joan catch Jo? 
20100304 

From Rusty: If Jo left his house at 6:30am traveling at 5mph and Joan left the same house 7:00am traveling at 8mph, at what time would Joan catch Jo? Answered by Penny Nom. 





A related rates problem 
20100303 

From Amanda: A circle is inscribed in a square. The circumference of the circle is increasing at a rate of 6 inches per second. As the circle expands, the square expands to maintain the tangency. Determine the rate at which the area of the region between the circle and square is changing at the moment when the cricle has an area of 25(pi) square inches. Answered by Penny Nom. 





15 men can do a piece of work in 7 days 
20100220 

From Kenneth: If 63 books cost $126, what will 125 books cost?
If 15 men can do a piece of work in 7 days, in how many days can 21 men do the same work? Answered by Penny Nom. 





Melting snow 
20100211 

From Cyndi: How long will it take for 5 feet of snow in an area of 17 by 12 to melt
at 35 degrees and cloudy. Answered by Robert Dawson. 





A decrease in travel time 
20100130 

From Mike: if a 100 mile trip averages 50 miles per hour, how much distance does one need to travel at 90 miles per hour to decrease travel time by 10% Answered by Stephen La Rocque. 





Growth rates 
20100126 

From Bhavya: Dear Sir/Ma'am,
I read in the text book that the growth rates of these 3 functions are as below
n^{2/3} < n/lg n < n^{0.99}
I tried the substitution method to check the correctness of this. But it gets really tough as n increase.
Is there any simpler way to understand the correctness of these growth rates?
Thanks and Regards,
Bhavya Answered by Robert Dawson. 





How long will it take to go 1 mile at 80 mph? 
20100119 

From Susann: How long will it take to go 1 mile at 80 mph? Answered by Penny Nom. 





Related Rates Problem 
20100112 

From Neven: A woman raises a bucket of cement to a platform 40 ft
above her head by means of a rope 80 ft long that passes
over a pulley on the platform. If she holds her end of
the rope firmly at head level and walks away at 5ft/s,
how fast is the bucket rising when she is 30 ft away
from the spot directly below the pulley?
(G. F. Simmons, Calculus with Analytic Geometry, pg.142) Answered by Penny Nom. 





How fast am I walking? 
20100110 

From KEVIN: IF I WALK 3 MILES IN 28 MINUTES, HOW FAST AM I WALKING? Answered by Penny Nom. 





Gail and Bill drove to the beach 
20100110 

From Nicole: Gail and Bill drove to the beach at an average speed of 50mph. They return home over the same road at an average speed of 55mph. The tip took 30 min less time. How far is the beach from their house? Answered by Stephen La Rocque. 





Related Rates of a Cylinderical Trough with a Horizontal Axis 
20091226 

From Emily: A cylinder is lying on it's side and being filled with water at a constant rate. Let the current height of water be t=0. When t=4, the cylinder is half full. When t=12, the cylinder is completely full. When is the rate of the height change increasing? Answered by Janice Cotcher. 





A pile of sand 
20091216 

From Malik: Sand is leaking out of a hole at the bottom of a container at a rate of 90cm3/min. As it leaks out, it forms a pile in the shape of a right circular cone whose base is 30cm below the bottom of the container. The base radius is increasing at a rate of 6mm/min. If, at the instant that 600cm3 have leaked out, the radius is 12cm, find the amount of leakage when the pile touches the bottom of the container. Answered by Harley Weston. 





... the original speed of the car was? 
20091208 

From venkatesh: A car travels a distance of 840 miles at a uniform speed if the speed of the car is increased 10 m/h more ,it takes two hours less to cover the same distance ,the original speed of the car was?
options:75 mh,65 m/h,60 m/h,55 m/h,45 m/h . Answered by Claude Tardif. 





How fast is the distance between the two cars decreasing? 
20091208 

From Jenny: Two cares are on a collision course toward point P. The paths of the two cars make a 30 degree angle with each other. The first car is 40 km from P, and traveling toward P at 16 km/hour. The second car is 50 km from P, traveling at 20 km/hour. How fast is the (straight line) distance between the two cars decreasing. (Hint: Law of Cosines) Answered by Harley Weston. 





Walking at 4/5 of his usual speed 
20091205 

From venkatesh: walking at 4/5 of his usual speed a man is 10 minutes too late .find his usual time. Answered by Penny Nom. 





A rate of 67 per 1000 
20091110 

From ANTHNONY: if the rate is 67 per 1000 the fraction is 67/1000 how do I present the expanded form Answered by Penny Nom. 





At what rate are the people moving apart? 
20091101 

From saira: A man starts walking north at 4 ft/s from a point P. 5 minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minute after the woman starts walking ? Answered by Harley Weston. 





How far will it travel in 45 mins? 
20091027 

From maison: I'm stuck on the formula i dont remeber what it is can you help me out
a car is traveling 80km/h how far will it travel in 45 mins Answered by Robert Dawson. 





Two jets are traveling towards each other 
20091005 

From jake: two jets are traveling towards each other and are 4000 km apart. the rate of one jet is 100 km/h faster than the rate of the other. if the jets pass each other after 2.5 hrs what is the rate of speed of the faster jet? Answered by Penny Nom. 





How long (distance)did it take to stop? 
20090930 

From mary: a car travelling at 50km/h stops in 4 seconds.how long (distance)did it take to stop? Answered by Harley Weston. 





How long would it take the students to arrive at their campsite 
20090918 

From amanda: ..."A group of 5th grade student in New Zealand are going camping. They will hike from Wellingto to Ruapehu. Then they will follow a trail for another 1/2 mile to their campsite." Ok, 1 mile equals 5,280 feet. The class decided one step length is equal to 15 inches and there are 85 steps in 1 minute. Now for the question: About how long would it take the students to arrive at their campsite, if they don't make any stops??? Answered by Robert Dawson. 





Paddling downstream 
20090904 

From Nicole: You head downstream on a river in a canoe. You can paddle at 6 km/h and the river is flowing at 2.4 km/h. How far downstream will you be in 55 minutes? Answered by Penny Nom. 





The distance traveled on a bike 
20090828 

From Dennis: I would like to calculate the distance traveled on a bike based on the size of the tire and on the number of revolutions per minute. This is based on the rider traveling at a constant speed. Answered by Stephen La Rocque. 





Two cars 
20090809 

From Jayna: If a car was traveling 20mph east and another car was traveling 30mph west,
assuming they were 300 miles apart, at what point would they collide at the center? Answered by Penny Nom. 





Susan's RRSP 
20090805 

From Polly: Susan contributed $500 every 6 months for 14 years into a RRSP earning interest @ 7.5% compounded semiannually. Seven years after the last contribution Susan converted the RRSP into a RRIF which is to pay her equal 1/4 payments for 16 years. If the first payment is due 3 months after the conversion into the RRIF and the interest on the RRIF is 9% compounded 1/4, how much will Susan receive every 3 months? Answered by Stephen La Rocque. 





A boat goes up stream 30 miles and ... 
20090629 

From Mohsin: A boat goes up stream 30 miles and down stream 44 miles in 10 hours. Again it goes up stream 40 miles and down stream
55 miles in 13 hours. Find the rates of the stream and of the boat.
I am confused and unable to get an answer. Please help.
Thanks Answered by Robert Dawson. 





Investing in multiple accounts 
20090626 

From Kenneth: Hello:
If an investor has $1000.00 to invest in multiple accounts, and he
wants a total return of 4%, is there one calculation that can be used
to determine what these amounts could be even though there may be
numerous amounts used as answers for most of the following examples?
For example,
Invest $1000.00 @ 2% and 5% for total return of 4%.
Invest $1000.00 @ 2%, 3% and 5% for total return of 4%.
Invest $1000.00 @ 2%, 3%, and 5% for total return of 4%.
Invest $1000.00 @ 2%, 3%, 4% and 5% for total return of 4%.
etc. Answered by Robert Dawson. 





A train crosses a tunnel 
20090625 

From Sabrina: A goods train of length one and half kilometres, is moving at a speed of 27km/hr. find the time taken by it to cross a tunnel of length seven and half kilometres Answered by Penny Nom. 





Mowing 17 lawns 
20090618 

From kevin: I need help setting up a equation to solve the following question: If a gardener can mow 3 lawns in 7 hours, how long should it take him to mow 17 lawns. I can solve the problem, but I don't know how to set up the equation. Answered by Penny Nom. 





The integral of x^x 
20090618 

From ANGIKAR: what would be the integration of (X^Xdx)?
give answer in details. Answered by Robert Dawson and Harley Weston. 





What was her rate of speed? 
20090506 

From Stephanie: beth paddled a canoe to a picnic spot in the river in 2 hours. she traveled up the stream back to the original spot in three hours. the current was two miles per hour.
What was her rate of speed? Answered by Robert Dawson and Stephen La Rocque. 





Mixing drinks 
20090502 

From samantha: If Steven can mix 20 drinks in 5 minutes, Sue can mix 20 drinks in 10 minutes and Jack can mix 20 drinks in 15 minutes. How much time will it take all 3of them working together to mix the 20 drinks? Answered by Stephen La Rocque and Claude Tardif. 





A word problem involving y = mx + b 
20090424 

From Devon: I need to put the following question in to y=mx+b form
I rent a gym for $150.00 for 30 students. Another time I rent the gym for @270.00 for 70 students. I need to also find a fixed rate. Answered by Stephen La Rocque. 





Filling a container with water 
20090422 

From frank: With one house pipe it takes 8 hours to fill a container with water. How long will it take with 4 house pipes of the same size and the same water pressure and same container? Answered by Robert Dawson. 





Interest rate over two periods 
20090412 

From Kenneth: Hello:
How is the average determined for the following amount if it earns
compound interest?
Here are my calculations, but I'm not certain that they are correct:
Initial amount invested $500.00:
$500.00 @ 5% = $525.00.
$525.00 @ 2% = $514.50.
$14.50/$500.00 = 2.9% divided by 2 equals an average of 1.45%.
If this average, 1.45%, is correct, how can I use 5% and 2% to
determine the same average?
Is it possible?
I thank you for your reply. Answered by Penny Nom. 





The integral of the square root of the sine function 
20090407 

From Indrajit: how to integrate this derivative???
∫√sinx Answered by Harley Weston. 





Sand falls from a conveyor belt 
20090401 

From Tracy: Sand falls from a conveyor belt at the rate of 10 cubic feet per minute onto a conical pile. The radius of the base is always equal to half the pile's height. How fast is the height growing when the pile is 5ft high? Answered by Stephen La Rocque. 





A spherical Tootsie Roll Pop 
20090401 

From Tracy: A spherical Tootsie Roll Pop you are sucking on is giving up volume at a steady rate of .8 ml/min. How fast will the radius be decreasing when the Tootsie Roll Pop is 20 mm across? Answered by Harley Weston. 





A 5 ounce object traveling at 60 mph 
20090327 

From James: if a 5 ounce object traveling at 60 mph at 100 feet above the ground, how far will it travel forward before hitting the ground. Answered by Robert Dawson. 





The rate of change of the volume of a sphere 
20090325 

From Kaylin: why the rate of change of volume of a sphere is not constant even though dr/dt is constant? Answered by Walter Whiteley. 





Related rates 
20090314 

From Jeevitha: The side of an equilateral triangle decreases at the rate of 2 cm/s.
At what rate is the area decreasing when the area is 100cm^2? Answered by Stephen La Rocque. 





Water drains from a conical tank 
20090311 

From Tyler: Water drains from a conical tank at the rate of 5ft/min^3. If the initial radius of the tank is 4' and the initial height is 10'.
a) What is the relation between the variables h and r? (height and radius)
b) How fast is the water level dropping when h=6'?
Thanks for the help, i'm stumped. Answered by Penny Nom. 





Related rates 
20090309 

From Megan: A plane flying with a constant speed of 330 km/h passes over a ground radar station at an altitude of 3 km and climbs at an angle of 30°. At what rate is the distance from the plane to the radar station increasing a minute later? Answered by Harley Weston. 





Two racers going around a track 
20090226 

From larisa: One racer makes a complete trip around the track every 28 seconds, while the other racer completes a trip around the track every minute. How many times will the first racer pass the second racer in 30 minutes? Answered by Robert Dawson. 





Amount of Water 
20090223 

From Jeanette: I received a water bill for 660,000 gal. of water. I don't think it is possible to pass this much water through a 1 1/2 inch pipe in 30 days. Is it possible to send that much water? I am assuming that there is a water pressure of 50 gpm, but I am not sure of that. It is a rural water system. Answered by Janice Cotcher & Robert Dawson. 





Fred in his canoe 
20090221 

From Susan: Fred paddled for 4 hours with a 7 km/hr current to reach a campsite. The return trip against the same current took 9 hr. Find the speed of Fred's canoe in still water. (Remember to include correct units) Answered by Penny Nom. 





Three people working in pairs 
20090221 

From bevaz: A and B can together do a piece of work in 6 days, B and C together in 20 days and C and A in 7.5 days. how long will they require separately for the work? Answered by Penny Nom. 





How long will it take Jake to catch up? 
20090220 

From Rebecca: John is walking 2 miles per hour. Jake leaves home 15 minutes later and is walking 5 miles per hour . How long will it take Jake to catch up ? Answered by Penny Nom. 





Joggong in opposite directions 
20090219 

From Mallory: Emma and her mother jog along a milelong circular path in opposite directions. They begin at the same place and time. Emma jogs at a pace of 4 mi/h, and her mother runs at 6 mi/h. In how many minutes will they meet? Answered by Robert Dawson. 





Rates 
20090210 

From Jennifer: write each phrase in simplest form
30oz in 24 gl
48leaves on 9 plants Answered by Harley Weston. 





Catching up to the car ahead 
20090206 

From Ashley: Car A passes Car B on a freeway. The driver of Car B decides to speed up to 75 mph and catch Car A, whose driver is maintaining a constant speed.
Looking ahead, the driver of Car B sees Car A go under an overpass oneqarter mile up the road. Exactly six minutes later, Car B catches Car A.
How fast was the car going? Answered by Penny Nom. 





How long will it take to make 50 miles? 
20090126 

From Tamara: An object is traveling at a speed of 45 km/minute. How long will it take to make 50 miles? Answered by Robert Dawson and Stephen La Rocque. 





How fast is the visible surface of the earth decreasing? 
20090124 

From Ray: A dive bomber loss altitude at a rate of 400 mph. How fast is the visible surface of the earth decreasing when the bomber is one mile high? Answered by Harley Weston. 





Water flowing from a cone to a cylinder 
20090123 

From Ray: Water is passing through a conical filter 24 cm deep and 16 cm across the top into a cylindrical container of radius 6 cm. At what rate is the level of water in the cylinder rising when the depth of the water in the filter is 12 cm its level and is falling at the rate of 1 cm/min? Answered by Harley Weston. 





In the shadow of a flagpole 
20090122 

From La: How fast is the length of the shadow of an 18 foot flagpole growing when the angle of elevation of the sun is 45 degrees and is decreasing at a rate of 10 degrees per hour? Answered by Harley Weston. 





A tortoise and a hare 
20090120 

From jenna: A tortoise can run with a speed of 0.14 m/s, and a hare can run 20 times as fast. In a race, they both start at the same time, but the hare stops to rest for 1.0 minutes. The tortoise wins by a shell (25 cm).
(a) How long does the race take?
(b) What is the length of the race? Answered by Stephen La Rocque. 





Lita and Rose start jogging on a 110 m circular track 
20090113 

From Narcisa: Lita and Rose start jogging on a 110 m circular track. They begin at the same point, but jog in opposite directions, lita at 8/3 meters per second and rose at 7/3 m per second. Find the number of times they they will pass each other during the first 15 minutes of jogging? Answered by Robert Dawson. 





Negative rate of change 
20090112 

From hemanshu: when i have to find rate of change of decrease in any value my ans comes in negative why?????????? Answered by Penny Nom. 





Related rates 
20081126 

From Lyudmyla: How fast is the volume of a cone increasing when the radius of its base is 2 cm and growing at a rate of 0.4 cm/s, and its height is 5 cm and growing at a rate of 0.1 cm/s? Answered by Harley Weston. 





How fast is the length of his shadow changing? 
20081122 

From Desiree: A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 2.3 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? Answered by Harley Weston. 





A conical funnel 
20081112 

From Rachael: Hello, I am a 10th grader in AP Calc, and can not figure out this question:
Water is running out of a conical funnel at the rate of 1 inch^3/sec. If the radius of the base of the funnel is 4 in. and the altitude is 8 in., find the rate at which the water level is dropping when it is 2 in. from the top. Answered by Harley Weston. 





Filling a tank with 2 taps 
20081110 

From Murray: 2 taps turned on together can fill a tank in 15 minutes. By themselves, one
takes 16 minutes longer than the other to fill the tank.Find the time taken to
fill the tank by each tap on it's own. Answered by Penny Nom and Victoria West. 





How fast is the distance between the airplanes decreasing? 
20081110 

From Crystal: At a certain instant, airplane A is flying a level course at 500 mph. At the same time, airplane B is straight above airplane A and flying at the rate of 700 mph. On a course that intercepts A's course at a point C that is 4 miles from B and 2 miles from A. At the instant in question, how fast is the distance between the airplanes decreasing? Answered by Harley Weston. 





The speed of the boat and the river 
20081028 

From Jenn: On the Amazon River in Brazil, a boat goes 50 miles upstream in 3 hours, later the boat returns to the starting point in 2 hours. What is the speed of the boat in still water and what is the current speed in this part of the Amazon? Answered by Penny Nom. 





Water is leaking from a conical tank 
20081024 

From Kimberly: Water is leaking out of an inverted conical tank at a rate of 12000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. Answered by Stephen La Rocque. 





Melting ice on a hemisphere 
20081020 

From heather: The top of a silo is the shape of a hemishere of diameter 20 ft. if it is coated uniformly with a layer of ice, and if the thickness is decreasing at a rate of 1/4 in/hr, how fast is the volume of ice changing when the ice is 2 inches thick? Answered by Penny Nom. 





Travelling home o the train 
20081018 

From nadha: MELISSA reached the railway station at 9.30am.The distance between the railway station and her hometown was 720 km.At what time would she reach her hometown if the train travelled at an average speed of 120 km/h? Answered by Penny Nom. 





If you are driving 40mph, how long does it take to go 2.5 miles? 
20081018 

From Nathan: If you are driving 40mph, how long does it take to go 2.5 miles? Answered by Penny Nom. 





Two planes flying in opposite directions 
20081016 

From samantha: i don't understand how to do this problem.two jets leave denver at 9:00 am one flying at a speed 50 km/h greater than the other, which is traveling west. at 11:00 am the planes are 2500 km apart. find their speeds. Answered by Penny Nom. 





Related rates 
20081016 

From Gisela: As sand leaks out of a hole in a container, it forms a conical pile whose
altitude is always the same as its radius. If the height of the pile is increasing
at a rate of 6 in/min, find the rate at which the sand is leaking out when the
altitude is 10in. Answered by Penny Nom. 





The rate of change of the volume of a cone 
20081015 

From Barbara: Suppose that both the radius r and height h of a circular cone change at a rate of 2 cm/s.
How fast is the volume of the cone increasing when r = 10 and h = 20? Answered by Harley Weston. 





The average rate of change of gasoline used 
20081006 

From JHulie: What is the average rate of change of gasoline used, measured in miles per gallons
if you travel 212 miles, then you fill your gas tank up again and it takes 10.8 gallons.
If you designate your change in distance as 212 miles and your change in gallons as 10.8? Answered by Penny Nom. 





Pumping water into a tank 
20080913 

From nadha: A water pump can fill a tank at the rate of 750 litres per hour.A tank has a capacity of
9000 litres. Find the time needed to fill 7/15 of the tank? Answered by Stephen La Rocque. 





Distance, time and rate 
20080911 

From Pauline: During the first part of a trip, a canoeist travels 100 miles at a certain speed. The canoeist travels 25 miles on the second part of the trip at a speed 5 mph slower. The total time for the trip is 5 hrs. What was the speed on each part of the trip? Answered by Penny Nom. 





Trough Filling with Water 
20080821 

From lanny: a triangular trough is 10 feet long, 6 feet across the top, and 3 feet deep. if water flows at the rate of 12 cubic inches per minute, find how fast the surface is rising when the water is 6 inches deep. Answered by Janice Cotcher. 





Two trains 
20080707 

From Elisha: Trains at each end of the 50. km long Eurotunnel under the English Channel start at the same time into the tunnel. Find their speeds if the train from France travels 8. km/h faster than the train from England and they pass in 17 minutes. Answered by Janice Cotcher and Penny Nom. 





A boat travels 20 kms upstream in 6 hrs ... 
20080707 

From vidya: A boat travels 20 kms upstream in 6 hrs and 18 kms downstream in 4 hrs. Find the speed of the boat in still water and the speed of the water current? Answered by Penny Nom. 





Slope and rate of change 
20080623 

From Lee: What is the difference between a slope and a rate of change? Answered by Stephen La Rocque. 





How far from each other will they be? 
20080618 

From kerstin: a women jogging at 6 mi/h passes a men biking in the opposite direction at 12 mi/h. if the maintain their speeds, how far from each other will they be 10 minutes after passing? Answered by Stephen La Rocque. 





The rate of change in the depth of the water 
20080612 

From Liz: A rectangular pool 50ft long and 30ft. wide has a depth of 8 ft. for the first 20 ft. for its length and a depth of 3 ft. on the last 20ft. of its length and tapers linearly for the 10 ft in the middle of its length. the pool is being filled with water at the rate of 3ftcubed/ min
at what rate is the depth of the water in the pool increasing after 15 hours? Answered by Harley Weston. 





Rate reduction 
20080611 

From Brian: we had a 2.43% scrap rate last year, compared to a 1.28% this year. does this come out to a .47% improvement?
Thanks for your help Answered by Harley Weston. 





The volume of a solid 
20080604 

From Tom: The base of a certain solid is a circle of radius 2 while cross sections perpendicular to the
base are isosceles triangles of height 1. What is the volume of the solid? Answered by Harley Weston. 





The length of a shadow 
20080527 

From Simon: A figure skater is directly beneath a spotlight 10 m above the ice. IF she skates away from the light at a rate of 6m/s and the spot follows her, how fast is her shadow's head moving when she is 8m from her starting point? The skater is (almost) 1.6m tall with her skates on. Answered by Stephen La Rocque and Harley Weston. 





A cat is walking on the tread of a tank 
20080526 

From Kathleen: If a cat is walking on the tread of a tank travelling at 28kmh how fast does the cat have to walk to avoid falling under the tracks? Answered by Janice Cotcher. 





Liquid is being pored into the top of a funnel 
20080525 

From Stella: Liquid is being pored into the top of a funnel at a steady rate of 200cm^3/s. The funnel is in the shape of an inverted right circular cone with a radius equal to its height. It has a small hole in the bottom where the liquid is flowing out at a rate of 20cm^3/s. How fast is the height of the liquid changing when the liquid in the funnel is 15cm deep?
At the instance when the height of the liquid is 25cm, the funnel becomes clogged at the bottom and no mo re liquid flows out. How fast does the height of the liquid change just after this occurs? Answered by Stephen La Rocque. 





55 mi/h for 4 h 
20080521 

From brenda: mary drove her car at 55 mi/h for 4 h. Use the formula d = rt (where d is the distance traveled, r is the rate of travel and t is the time traveled) to determine how far mary traveled. Answered by Penny Nom. 





How much seed do I need per plot? 
20080513 

From Jaye: The plot size is 5 m X 2 m. The seeding rate is 12Kg/ha. How much seed do I need per plot? I have failed to get a proper answer.
Thank you. Answered by Penny Nom. 





A crate is launched from an airplane 
20080510 

From Michelle: A crate is launched from an airplane, 75 feet above the ocean, with an initial vertical velocity of 60 feet per second. How many seconds will it take for the crate to hit the water? Round to the nearest hundredth. Answered by Harley Weston. 





How far is it from the top to the bottom of the mountain? 
20080509 

From Matthew: A ski lift carries a skier up a slope at a rate of 2 miles per hour. Following the same path down the mountain, the hiker has a rate of 4 miles per hour. If the round trip took 3 hours, how far is it from the top to the bottom of the mountain? Answered by Penny Nom. 





5 miles per hour for 30 minutes 
20080429 

From Joshua: A boat travels at a constant rate of 5 miles per hour for 30 minutes. How far does the boat go in that time? Answered by Penny Nom. 





Working together 
20080426 

From Joanna: If 8 men took 15 days to paint a building . How many more men are needed to paint the building in 6 days ? Answered by Stephen La Rocque. 





Related rates 
20080425 

From Mary: A rectangular box is 10 inches high. It's length increases at a rate of 2 inches per second and it's width decreases at the rate of 4 inches per second. When the length is 8 inches and the width is 6 inches, what is the rate of change of the volume? Answered by Stephen La Rocque. 





Driving home from work 
20080425 

From brittanygrant: brittany house is 37 miles from his job. He leaves at 6:25pm, and drives 49 miles per hour. what time will he get beck home? Answered by Penny Nom. 





What is the altitude of the plane? 
20080423 

From tiana: If a plane is taking off and its climbing at 380 feet per minute at 205 MPH, what is the altitude of the plane after it traveled 115 miles? Answered by Penny Nom. 





What is the integral of 13sin^3(x)*cos^7(x)dx? 
20080422 

From Cathrine: I am having trouble integrating this problem. It says to evaluate the integral but I don't know what to do or how to do it.
It is the integral of
13sin^3(x)*cos^7(x)d Answered by Harley Weston. 





45 miles at 50 mph 
20080418 

From Willie: If you travel at 50 mph, how long does it take to go 45 miles? Answered by Penny Nom. 





Jill and Matt run on a 400 meter long oval track 
20080411 

From jamie: Jill and Matt run on a 400 meter long oval track. Jill and Matt each run 10 laps daily and both run in the same lane.
One day Jill and Matt began at the same point, but ran around the track in opposite directions. Jill ran at a constant speed that was 2 meters per second faster than matts constant speed. Jill passed Matt for the first time in 40 seconds. Jill ran at a constant rate of how many meters per second? Answered by Penny Nom. 





Two trains left the city at the same time 
20080411 

From J: Two trains left the city at the same time, traveling in opposite directions. The eastbound train travels for ten hours and the westbound train travels for 5 hours. They are now 1300 km apart. The westbound train's rate is 20 km/hr faster than the eastbound train. Find the speed of each train. Answered by Penny Nom. 





A river boat 
20080409 

From jessica: heres a crazy riddle a river boat that travels 12 mph in still water makes a
pleasure trip to a city upstream and back in 3 hours. If the city is 10 miles aways, what is the rate of the current Answered by Stephen La Rocque. 





The average rate of change 
20080329 

From Tom: For the function x/3x1 find the average rate of change between the interval x=1 and x=5? Answered by Harley Weston. 





What was the annual interest rate? 
20080318 

From Javier: A business invests $10,000 in a savings account for two years. At the beginning of the second year, and additional $3,500 is invested. At the end of the second year, the account balance is $15,569.75. What was the annual interest rate? Answered by Penny Nom. 





A train and a boat 
20080315 

From Sabrina: A railroad bridge is 20m above, and at right angles to, a river. A person in a train travelling at 60 km/h passes over the centre of the bridge at the same instant that a person in a motorboat travelling at 20km/h passes under the centre of the bridge. How fast are the two people separating 10s later? Answered by Harley Weston. 





Average speed up and down a hill 
20080309 

From Judith: An old brokendown ar has to travel a two miles route, up and down a hill.
Because it is so old, the car travels the first mile  the ascent  at an average
speed of fifteen miles per hour. How fast must it cover the second mile 
the decent  to achieve an average speed of thirty miles per hours?
(There is a missing variable, but what is it?) Answered by Penny Nom. 





A package of supplies is dropped from a plane 
20080229 

From beth: An airplane flies west at a speed of 200m/s. A package of supplies is dropped from the plane to some campers. if the plane is at an altitude of 1000meters, how far from the campers should the package be dropped to land near them? ignore wind resistance. Answered by Stephen La Rocque. 





It takes Mark 3 hours to paint a picture 
20080227 

From Suzanne: If Mark takes 3 hours to paint a picture and it takes Henry three times as
long to paint the same picture, then:
How many pictures can each paint in 10 hours?
How many pictures can they paint together in 10hours? Answered by Penny Nom. 





Integrate (4+x^2) / (x (2+x)^2) 
20080225 

From Russell: integrate (4+x^2) / (x (2+x)^2)
I really need to know the steps involved in doing this integral, I already have
the answer through wolfram integrator but for some reason can not
get to the answer given. Please help Answered by Harley Weston. 





Growth factor and growth rate 
20080213 

From william: what is the difference between growth factor and growth rate? Answered by Stephen La Rocque and Harley Weston. 





Two job offers 
20080210 

From Kelly: Raja has been offered two jobs.
Each of these jobs takes 24 weeks to complete.
One job pays $3440 every 8 weeks. The other job pays $2700 every 6 weeks.
Raja wants to accept the job that pays more per week.
Show how to use equations to help Raja make her choice. Answered by Penny Nom. 





Two cars are on a collision course 
20080206 

From Danielle: Two cars are on a collision course, heading straight towards each other. One car is traveling 45 miles per hour and the other is traveling 75 miles per hour. How far apart will the two cars be one minute before they collide. Answered by Penny Nom. 





How fast does each travel? 
20080205 

From kRYSTAL: May rides her bike the same distance that leah walks. May rides her bike 10km/h faster than leah walks.
Its takes May 1 hour and Leah 3 hours to travel the sam distance, how fast does each travel? Answered by Penny Nom. 





Driving from Mathville to Bioville 
20080201 

From Ariel: Mahtab took hours to drive from Mathville to Bioville, a total of 485 km.
She drove part by highway at 80 km/h and part by slow roads at 30 km/h.
How many hours did she drive each type of road? Answered by Penny Nom. 





2 km at 5.00 m/s 
20080129 

From Stephanie: How much time does it take for a student running at an average speed of 5.00 m/s to cover a distance of 2.00 km? Answered by Penny Nom. 





The integral of Sqrt(sin x) 
20080124 

From tariq: find integration
Sqrt(sinx) dx , from x=bi/4 to 3bi/4 Answered by Harley Weston. 





Integration 
20080123 

From Russell: how to integrate
1/x ln^2 [x] dx Answered by Harley Weston. 





An integral 
20080123 

From Russell: I checked out the integration site, but would like an explanation for integrating this integral
Tan^3[x] * _dx__
Cos^2[x]
Hope this input is correct!
the part that throws me off is Tan^3[x], if it was squared rather than cubed I would have a easier time
dealing with this one. Answered by Harley Weston. 





Two bikers race on a circular track 
20080117 

From Abby: Two bikers race on a circular track. Biker A can circle the track in 8 minutes, and Biker B can circle the track in 6 minutes. From the start of the race, how many minutes will it be before Biker B overtakes Biker A? Answered by Stephen La Rocque and Penny Nom. 





Driving towards each other 
20080116 

From Shanda: Paul and Charlene are 420 miles apart. They start toward each other with Paul driving 16 miles per hour faster than Charlene. They meet in 5 hours. Find Charlene's speed. Answered by Stephen La Rocque. 





How fast does Kim paddle in still water? 
20080107 

From Fran: Kim paddled a canoe 10 km upstream and then paddled back to his starting point. If the rate of the current was 2 km/h that day and the whole trip took 3.75 h, how fast does Kim paddle in still water? Answered by Penny Nom. 





A spherical bubble gum bubble 
20071231 

From Houston: Bazooka Joe is blowing a spherical bubble gum bubble. Let V be the volume in the bubble, R the inside of the bubble, and T the thickness of the bubble. V, T, and R are functions of time t.
(a) Write a formula for V in terms of T and R. Hint: Draw a picture
(b) Assume that the amount of bubble gum in the bubble is not changing. What is V'(t)?
(c) After 5 minutes of blowing a bubble gum bubble, the bubble is 3ft in diameter and .01 feet thick. If the inside radius of the bubble is expanding at a rate of .5 feet per minute, how fast is the thickness changing? Hint: Remember that the volume of gum in the bubble does not change over time. Answered by Harley Weston. 





Sally, Charlie and Snoopy in a race 
20071229 

From Marie: Sally and Charlie are having a race. Charlie gets a 900 foot lead and runs 8
inches per second. Sally begins at the starting line and runs at a rate of 5 ft per
second. Charlie's dog is also in the race. Snoopy starts 1100 feet ahead of Sally and runs
toward the starting line at a rate of 1 foot per second.
When will Charlie and Snoopy past each other and how far will they be from the starting line? Answered by Penny Nom. 





Related rates  tree growth 
20071209 

From Christy: How do I go about answering this question, I know I have to find dv/dt, but I'm not sure how to start.
The volume of a certain tree is given by V= 1/12pie C^2h where C is the circumference of the tree at the ground level and h is the height of the tree. If C=5feet and growing at the rate of 0.2feet per yer, and if h=22feet and is growing at 4 feet per year, find the rate of growth of the volume, V. Answered by Stephen La Rocque and Harley Weston. 





20 gallons of concentrate 
20071120 

From Ken: What is the formula I need to use?
I have 20 gallons of concentrate. I am mixing it at 0.5%.
How many total gallons of solution will I end up with?
I came up with 4000 gallons of solution  but not using the easiest of methods. Answered by Penny Nom. 





Red hens and white hens 
20071118 

From Rapin: In 15 days, 4 white hens and 3 red hens lay eggs equal to 3 white hens
and 5 red hens lay eggs in 12 days , how many days that white hens can
lay eggs the same amount as red hens lay eggs in 20 days ? Answered by Steve La Rocque and Penny Nom. 





Our company charges a 4% margin 
20071113 

From Nadja: Our company charges a 4% margin on top of a pay rate to obtain the total charge rate. A
client is denying that the total charge rate is calculated in the following
way:
Pay rate/0.96 = Charge rate
Please could you provide me with an explanation which I can pass on as
to why it is calculated in this way? Answered by Penny Nom. 





Related Rates (streetlamp and shadow) 
20071109 

From Casey: A street light is mounted at the top of a 15ft pole. A man 6ft tall walks away from the pole at a rate of 5ft per second. How fast is the tip of his shadow moving when he is 40ft from the pole? Answered by Stephen La Rocque and Penny Nom. 





Simple interest 
20071108 

From Lee: If a bank pays 3% simple interest anually on savings, and you did not take any money out of your account, how much money would you have deposited to earn $45 in interest Answered by Stephen La Rocque. 





Related Rates (a water trough) 
20071107 

From Christina: A rectangular trough is 3ft long , 2ft across the top and 4 ft deep. If water flows in at the rate of 2ft^3/min, how fast is the surface rising when the water is 1 ft deep ? Answered by Stephen La Rocque. 





The hourly rate plus time and a half 
20071105 

From Barbara: simon arrive to work at 8:15am  and left at 10:30pm if simon gets paid by the hour at a rate of $10.00 and time 1/2 for any hours worked over 8 in a day how much did he get paid?
a.120.25
b. 160.75
c. 173.75
d. 180
e. 182.50 Answered by Penny Nom. 





Two cars 
20071104 

From Jalisa: If Car "A" is traveling 50 miles per hour and car "B" is traveling 60 miles per hour, how long will it take car "B" to catch up to car "A" if car "A" had a 15 mile head start. Answered by Penny Nom. 





The rate of change of the concentration of a solution 
20071030 

From Nicholas: A barrel initially has two kg of salt dissolved in twenty liters of water. If water flows in the rate of 0.4 liters per minute and the wellmixed salt water solution flow out at the same rate, how much salt is present after 8 minutes?
I tried working backwards given the answer but I can't seen to get their answer of ~1.7kg. Any help would be great! Thanks Answered by Harley Weston. 





How to solve related rates problems 
20071027 

From David: Can you plz explain how and where you come up with an equation to solve this?
Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2 centimeters per second. Answered by Stephen La Rocque. 





Related rates 
20071026 

From David: A trough is 12 feet long and 3 feet across the top.(look like an upsidedown triangle square). Its ends are isosceles triangles with altitudes of 3 feet.
a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when h is 1 foot deep?
b) If the water is rising at a rate of 3/8 inch per minute when h=2, determine the rate at which water is being pumped into the trough.
thank you so much for helping me out Answered by Stephen La Rocque. 





The rate of change of the area of a triangle 
20071022 

From Ahlee: So my question is:
The included angle of the two sides of a constant equal length s of an isosceles triangle is ϑ.
(a) Show that the area of the triangle is given by A=1/2s^2 sinϑ
(b) If ϑ is increasing at the rate of 1/2 radian per minute, find the rate of change of the area when ϑ=pi/6 and ϑ=pi/3.
(c) Explain why the rate of change of the area of a triangle is not constant even though dϑ/dt is constant Answered by Penny Nom. 





A rectangular trough 
20071018 

From David: A rectangular trough is 2 meter long, 0.5 meter across the top and 1 meter deep. At what rate must water be poured into the trough such that the depth of the water is increasing at 1m/min. when the depth of the water is 0.7m.
I know this involves implicit differentiation somehow, but the 3 variables, since V=l*w*h for a rectangle is confusing me. I'm not sure whether one of the variables should be fixed or not, since I'm not getting anywhere with this right now. Any help would be great. Answered by Stephen La Rocque and Penny Nom. 





A conical cup 
20071018 

From Nicholas: Water is leaking out of a small hole at the tip of a conical paper cup at the rate of 1cm^3/min. The cup has height 8cm and radius 6cm, and is initially full up to the top. Find the rate of change of the height of water in the cup when the cup just begins to leak.
Since V= (pi/3)r^2h, how do I eliminate a variable or change the equation so I that I can answer the question? Thanks. Answered by Penny Nom. 





Related rates 
20071015 

From Alexis: Example 1. An observer is tracking a small plane flying at an altitude of 5000 ft. The plane flies directly over the observer on a horizontal path at the fixed rate of 1000 ft/min. Find the rate of change of the distance from the plane to the observer when the plane has flown 12,000 feet after passing directly over the observer. Answered by Stephen La Rocque. 





The aggregate percentage for the year 
20071011 

From Lisa: I would like to add up twelve percentages then divide it by 12 to get the aggregate percentage for the year. How do I accomplish this task? Thanks. Answered by Penny Nom. 





The average rate of change of a function 
20071011 

From vern: Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(X)=sinX for the inverval [0,pi/6]? Answered by Harley Weston. 





How long was she be raking leaves? 
20071005 

From Skylar: A family is raking leaves. Betsy fills 3 bags to every 2 bags the Meri fill. After a while, Jim joins them. He fills 3 bags to every 2 bags that Betsy fills, but he only works half as long. When finished, they have completely filled 58 bags. If Meri takes 6 minutes to fill one bag, how long was she raking leaves? Answered by Chris Langdon. 





Hitting golf balls 
20071003 

From Jessi: Ben is out at the practice range hitting golf balls. How much further will a golf ball with an initial speed of 75.0 m/s go when projected at 45.0 degree than when projected at 30.0 degree? Answered by Stephen La Rocque. 





Will the stone catch up the parachute? 
20071003 

From anoi90: a small parachute dropped from a 30.0 m high cliff falls with a constant velocity of 1.20 m/s. 20.0 s after the parachute is dropped, a stone is dropped from a cliff. Will the stone catch up the parachute before it reaches the ground? (Include numerical proof!) Answered by Stephen La Rocque. 





Peter and Charles mow a lawn 
20070927 

From Nancy: Peter takes 15 minutes longer to mow the lawn by himself than Charles does. Together they can mow the lawn in 18 minutes. How long would it take Charles to mow the lawn by himself? Answered by Stephen La Rocque. 





Water flowing into a tank 
20070921 

From andrew: Hi, I've been having real trouble visualizing this problem as apposed to a conical tank.
It says the base of a pyramidshaped tank is a square with sides of length 12 feet. The
vertex of the pyramid is 10 feet above the base. The tank is filled to a depth of 4 feet, water is flowing
into the tank at the rate of 2 cubic feet per minute. Find the rate of change of the depth of water in the tank. Answered by Harley Weston. 





How many gallons per minute? 
20070912 

From Diane: HI
I have a natural water spring and I am trying to determine how many gallons per minute will flow in an 8" pipe? I know one gallon is 231 cubic inches and V=nr2h  so if i had one foot of 8" pipe it would hold 2.6 gallons but I'm looking for the flow rate of how many gallons per minute? Thanks for your help. Answered by Stephen la Rocque. 





How many miles per hour was I driving 
20070911 

From Gary: How many miles per hour was I driving to go 37 miles in 13 minutes Answered by Victoria West. 





Water in a conical tank 
20070910 

From Greg: Joe is conducting an experiment to study the rate of flow of water from a conical tank.
The dimensions of the conical tank are:
Radius at the initial water level = 13.7 cm
Radius at the reference point = 12.8 cm
Initially the tank is full of water. There is a circular orifice at the bottom of the conical
tank with a diameter of 0.635 cm. The water drains from the conical tank into an empty
cylindrical tank lying on its side with a radius of 0.500 ft and a length L (ft).
Joe observed the water discharged with an average velocity of 1.50 m/s as the water level
lowered from the initial height of 14.0 cm to 5.00 cm in the conical tank. Answer the
following:
1. If the initial height of water in the conical tank is 14.0 cm (measured from the
reference point, see Fig. 1), how long in seconds will it take for the water level to drain to
a height of 5.00 cm?? NOTE: Height refers to the vertical height.
What formula would I use to find out how long in seconds it takes for the water level to drop? Answered by Harley Weston. 





Ten workers perform one job in five days 
20070821 

From Kenneth: If ten workers perform one job in 5 days, one person performs one job in how many days?
Here is the calculation that I used:
(10 workers X 1 job X 5 days)/(1 person X 1 job X ? days)
The above equals 50/1, and the answer is 50 days because 50/1 = 50. In this calculation I can determine any number of workers or days if the number of jobs remains the same as that in the group of factors from the numerator (10 workers X 1 job X 5 days), that is 1 job. Here is another example to help clarify: (10 workers X 1 job X 5 days)/(? workers X 1 job X 10 days) This equals 50/10. The answer is 5 workers. So, if 10 workers can perform 1 job in 5 days, 5 workers can perform 1 job in 10 days.
Now, if I replace "1 job" from (10 workers X 1 job X 5 days) with a different number, for example, 4 jobs, this amount will prevent the calculation from producing the correct answer.
Here is an example: (10 workers X 1 job X 5 days)/(10 workers X 4 jobs X ? days) Mathematically, the calculation works, but the answer, 1.25 days, is not correct, if I'm not mistaken. If 10 workers can perform 1 job in 5 days, they cannot, by working at the same rate, perform 4 jobs in 1.25 days.
Can you explain, with a simple explanation, why the number representing the jobs in this calculation needs to be the same in the group of factors in both the numerator and in the denominator in order to provide the correct answer? Answered by Harley Weston. 





Three runners on a circular track 
20070804 

From Arul: A, B and C run around a circular track starting from the same point simultaneously and in the same direction at speeds of 4 kmph, 6 kmph and 8 kmph respectively. If the length of the track is 400 meters, when will A, B and C meet at the starting point for the first time after they started the race?
4 In the above question, when will they meet for the first time after they started the race? Answered by Penny Nom. 





At what speed does Chuck travel? 
20070727 

From Jessica: Chuck and Dana agree to meet in Chicage for the weekend. Chuck travels
224 miles in the same time that Dana travels 204 miles. If Chuck's rate of travel
is 5 mph more than Dana's, and they travel the same length of time, at what speed
does Chuck travel? Answered by Harley Weston. 





Comparing flow rates of two pipes 
20070714 

From Kenneth: If two water pipes are 3 feet long, but one of them has a 1 foot diameter and the second has a 1 1/2 foot diameter, what simple mathematical method can be used to determine how much faster one pipe can drain water than the other pipe? Answered by Stephen La Rocque. 





What is his regular hourly rate? 
20070612 

From Gilligan: John Ritter, who is paid timeandahalf for hours greater than 40, had a gross weekly wage of $442 for 48 hours. What is his regular hourly rate? Answered by Penny Nom. 





Rate of change of distance between the clock hands 
20070605 

From Jonathan: A certain Clock has a minute hand with a length of 4 inches long and an hour hand with a length of 3 inches long. How fast is the distance between tips of these hands changing at 9:00? Answered by Stephen La Rocque. 





Probability tree  two switches failing 
20070605 

From Maura: draw a probability to show the outcomes of two new switches (a) of both switches being faulty (b)both switches are not faulty (c) that one switch is faulty. The failure rate is 1/10. Answered by Stephen La Rocque. 





A circular blob of molasses 
20070528 

From Julie: A circular blob of molasses of uniform thickness has a volume of 1 m^3.
The thickness of the molasses is decreasing at a rate of 0.1 cm/hour.
At what rate is the radius of the molasses increasing when the radius is 8
m?
Thanks,
Julia Answered by Penny Nom. 





Constant rate of sand falling in a cone 
20070520 

From Nhi: Sand is falling into a conical pile . After 5 min. the pile has radius 24 and height 26 . After 7 min. tell how far the point c. is from the top of the cone (A). Answered by Stephen La Rocque. 





A growing heap of sand: related rates 
20070423 

From Charles: Sand falls on to a horizontal ground at the rate of 9m ^ 3 per second and forms a heap in the shape of a right circular cone with vertical angle 60. Show that 10 seconds after the sand begins to fall, the rate at which the radius of the pile is increasing is 3 ^ (1/3) * (4/pi) ^ (1/3) m per minute. Answered by Stephen La Rocque and Penny Nom. 





Liquid is being poured into the top of a funnel 
20070419 

From neroshan: Liquid is being poured into the top of a funnel at a steady rate of 200cm^3/s.
The funnel is in the shape of an inverted right circular cone with a radius
equal to its height. It has a small hole at the bottom where the liquid is
flowing out at a rate of 20 cm^3/s. How fast is the height of the liquid
changing when the liquid in the funnel is 15 cm deep?
At the instant when the height of the liquid is 25cm, the funnel becomes clogged
at the bottom and no more liquid flows out. How fast does the height of the
liquid change just after this occurs? Answered by Penny Nom. 





Lowest terms 
20070417 

From Dawn: For some reason, I have a hard time with Ratio and/or rate problems.
My problem is as follow: Write each rate or ratio as a fraction in lowest
terms. 6 days to 39 hours Answered by Stephen La Rocque. 





Write each rate or ratio as a fraction in lowest terms 
20070417 

From Dawn: Write each rate or ratio as a fraction in lowest terms.
$59.00 for 231 minutes Answered by Stephen La Rocque. 





Rent per square foot per month 
20070416 

From julie: my property is 60'x66'
rent is $700 a month
how much per inch per month does this equal
how much per foot per month does this equal Answered by Penny Nom. 





Water is being pumped into a trough 
20070409 

From Michael: Water is being pumped into a trough that is 4.5m long and has a cross section in the shape of an equilateral triangle 1.5m on a side. If the rate of inflow is 2 cubic meters per minute how fast is the water level rising when the water is 0.5m deep? Answered by Stephen La Rocque. 





A delivery driver travels from Belfast to an address in Dublin 
20070404 

From Gerry: A delivery driver travels from Belfast to an address in Dublin.The total distance fro the round trip
was x%.The time required for the forward trip was xx hours.Due to heavy traffic during the return trip an extra xx was required
How much slower was the delivery drivers speed on the return trip?.
I was wondering what equation would you use to solve this question. Answered by Penny Nom. 





Which compounding period will make the interest rate be as low as possible? 
20070327 

From lilly: if i decide to loan a friend 12000 and he said that he will pay it back in a single payment of 16000 after 5 years.
which compounding period will make the interest rate be as low as possible? daily or yearly?
how can i calculate each (yearly compound and daily compound)? Answered by Penny Nom. 





Cost of ribbon 
20070320 

From kelly: we are doing rate
Ribbon costs $1.44 for 3m
a). What is the cost per meter?
b). How much would 5 m of ribbon cost?
c). How much ribbon could you buy for $12.00 Answered by Penny Nom. 





Two busses 
20070315 

From Devon: A bus leaves a station at 1 p.m., traveling west at an average rate of 44 mi/h. One hour later a second bus leaves the same station, traveling east at a rate of 48 mi/h. At what time will the two buses be 274 mi apart? Answered by Steve La Rocque and Penny Nom. 





At what rate is the area of the triangle changing? 
20070224 

From mac: two sticks 3.5 feet long are hinged together and are stood up to form an isosceles triangle with the floor. The sticks slide apart, and at the moment when the triangle is equilateral, the angle is increasing at the rate of 1/3 radian/sec. At what rate is the area of the triangle increasing or decreasing at that moment? Mac Answered by Penny Nom. 





How long does she have to get out of the way? 
20070223 

From Sharon: How do I solve this? The height h (in feet) of an object that is dropped from the height of s feet is given by the formula h = s  16t^2 , where t is the time the object has been falling. A 6 foot tall woman on a sidewalk looks directly overhead and sees a window washer drop a bottle from the 6 story. How long does she have to get out of the way? Round to the nearest tenth. (A story is 12 feet.) Answered by Penny Nom. 





Water in a triangular trough 
20070130 

From Trina: the trough is 5 feet long and its vertical cross sections are inverted isosceles triangles with base 2 feet and height 3 feet. water is draining out of the trough at a rate of 2 cubic feet per minute. at any time t, let h be the depth and v be the volume of water in the trough. a. find the volume of water in the trough when it is full b. what is the rate of change in h at the instant when the trough is .25 full by volume? c. what is the rate of change in the area of the surface of the water at the instant when the trough is .25 full by volume? Answered by Penny Nom. 





A boat travels 60 miles downstream in the same time it takes to go 36 miles upstream 
20070127 

From Liz: A boat travels 60 miles downstream in the same time it takes to go 36 miles upstream. The speed of the boat in still water is 15 mi/h greater than the speed of the current. Find the speed of the current. Answered by Stephen La Rocque. 





Baby weight gain 
20070126 

From Reynaldo: A baby weighed 3.2 kg at birth. She gained 0.17 kg per week. How old was she when she weighed 5.75 kg? Answered by Penny Nom. 





A college student jogs from his home to the beach 
20070123 

From Garmani: A college student jogs from his home to the beach at 8 miles per hour. He visits with his friends on the beach for 4 hours, and then his friends drive him home at the rate of 40 miles per hour. If the student returns home after 6 hour, what is the distance from his home to the beach? Answered by Penny Nom. 





At what speeds were they travelling? 
20070116 

From Maria: Kim and Julie are joining Nicole at her parents cottage for the weekend. The cottage is 150km away from their neighbourhood. Kim can leave directly after school but Julie will be leaving after band practice, an hour and a quarter later. Kim took her time and drove slowly, averaging 20km/h slower than Julie. They both arrived at the same time. At what speeds were they travelling? Answered by Stephen La Rocque. 





What is his rate of walking for the entire trip? 
20070112 

From Garmani: A man walks 500 feet at 4 miles per hour and returns to the starting point at 3 miles per hour. What is his rate of walking for the entire trip? Answered by Stephen La Rocque. 





Integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 
20070109 

From James: How do you integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 Answered by Penny Nom. 





What is the speed of the current? 
20070107 

From Bob: A motorboat can maintain a constant speed of 16 miles per hour relative to the water. The boat makes a trip upstream to a certain point in 20 minutes; the return trip takes 15 minutes. What is the speed of the current? Answered by Haley Ess. 





Annualized 
20070103 

From Dan: What would my rate of compound interest be per month? Invested $80,000 and in 150 days have increased that amount by $14,300. Answered by Stephen La Rocque. 





Cary was going to meet Jane at the airport, ... 
20061211 

From Carmen: cary was going to meet jane at the airport, if he traveled 60mph, he would arrive one hour early, and he traveled 30 mph, he would arrive one hour late. how far was the airport? recall: distance=speed(time) Answered by Stephen La Rocque. 





Rate of a boat and a river 
20061209 

From Dale: If a boat goes downstream 72 miles in 3 hours and upstream 60 miles in 6 hour, the rate of the river and the rate of the boat in still water respectively are ____? Answered by Stephen La Rocque. 





A particle moving along a curve 
20061118 

From Rachel: a particle is moving along the curve whose equation is: (xy^3)/(1+y^2)=8/5 assume the xcoordinate is increasing at the rate of 6 units per second when the particle is at the point (1,2). a. at what rate is the ycoordinate of the point changing at that instant? b. is the particle rising or falling at that instant? Answered by Penny Nom. 





An aircraft and a missile 
20061118 

From Sarah: an aircraft is flying at a constant altitude with a constant speed of 600mph. an antiaircraft missile is fired on a straight line perpendicular to the flight path of the aircraft so that it will hit the aircraft at a point P. at that instant the aircraft is 2 miles from the impact point P the missile is 4 miles from P and flying at 1200 mph. at that instant, how rapidly is the distance between missile and aircraft decreasing? Answered by Stephen La Rocque. 





Wheat is poured on a conical pile 
20061117 

From Rachel: wheat is poured through a chute at the rate of 10 cubic feet per minute and falls in a conical pile whose bottom radius is always half the altitude. how fast will the circumference of the base be increasing when the pile is 8 feet high? Answered by Penny Nom. 





The rate of change of the perimeter of a square 
20061107 

From Karli: Find the rate of change of the perimeter of a square with respect to its area. Answered by Stephen La Rocque. 





A boat is being pulled towards a dock. 
20061106 

From Cassie: A boat is being pulled towards a dock. If the rope is being pulled in at 3 feet per second, how fast is the distance between the dock and the boat decreasing when it is 30 feet from the dock? Answered by Penny Nom. 





A melting snowball 
20061106 

From Jay: A snowball melts at a rate proportional to its surface area. Show that its radius shrinks at a constant rate. If it melts to 8/27 of its original volume in 20 minutes, how long will it take to melt completely? Please I need your help. Answered by Stephen La Rocque. 





Going from one town to another 
20061104 

From Austin: Going from one town to another,a man drives his car at 35 miles an hour and returning he drives at 25 miles an hour. the round trip takes 6 hours. Find the distance between the towns. Answered by Stephen La Rocque. 





Paddeling on the Alleman River 
20061031 

From Anita: Len is planning a threehour trip down the Alleman River and back to his starting point. He knows that he can paddle in still water at 3 mph and that the rate of the current is 2 mph. How much time can he spend going downstream? How far downstream can he travel? Answered by Penny Nom. 





Water is being pumped into the pool 
20061024 

From Jon: A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deeps at the deep end. Water is being pumped into the pool at 1/4 cubic meters per minute, an there is 1 meter of water at the deep end.
a) what percent of the pool is filled?
b) at what rate is the water level rising? Answered by Stephen La Rocque. 





How fast is the water level rising when the water is 1 meter deep? 
20061019 

From Don: The cross section of a 5meter trough is an isosceles trapezoid with a 2meter lower base, a 3meter upper base and an altitude of 2 meters. Water is running into the trough at a rate of 1 cubic meter per minute. How fast is the water level rising when the water is 1 meter deep? Answered by Stephen La Rocque. 





The velocity of a pendulum, part II 
20060907 

From Erin: We saw the question in your database about the velocity of a pendulum swinging.....It is the same exact question....but there is another question......it says....
"estimate the instantaneous rate of change of d with respect to t when t = 1.5. At this time, is the pendulum moving toward or away from the wall? Explain." Answered by Harley Weston. 





How fast is the water level rising 
20060812 

From Erin: Water runs into a conical tank at the rate of 9ft^{3}/min. The tank stands point down and has a height of 10 ft. and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft. deep? (V=1/3 pi r^{2} h). Answered by Penny Nom. 





I want to calibrate a glucose meter 
20060727 

From John: If I have 3 glucose control solutions having:
Low  50 mg/dl
Normal  100 mg/dl
High  330 mg/dl
I want to calibrate a glucose meter but I need more than these 3 data points. How can I mix these up to get different concentrations in between 50 and 100, 100 and 330. I would like to mix up different concentrations and put them in test tubes or whatever to do some calibration work. Like 70 mg/dl, 90 mg/dl, etc. More data points should be in between say 50 and 100 since this is normal, 90 mg/dl is considered close to normal.
Answered by Stephen La Rocque. 





If all of them work together, ... 
20060727 

From Kakron: Pipe A can fill in 20 mins and pipe B can fill in 30 mins and pipe C can empty the same in 40 mins. If all of them work together, find the time taken to fill the tank? Answered by Stephen La Rocque. 





What is the interest on $50,000.00 at 5% for 40 days? 
20060712 

From A borrower: What is the interest on $50,000.00 at 5% for 40 days? Answered by Penny Nom. 





Flying with and against the wind 
20060706 

From Tim: When a plane flies with the wind, it travels 800 miles in 4 hours. Against the wind it takes 5 hours to cover the same distance. Find the rate of the plane in still air and the rate of the wind. Answered by Stephen La Rocque. 





Painting a house 
20060628 

From A student: If Sally can paint a house in 4 hours, and John can paint the same house in 6 hour, how long will it take for both of them to paint the house together? Answered by Stephen La Rocque and Penny Nom. 





Rates and percentages 
20060620 

From Todd: If rates are about to rise 1.84%, from the current 5.30% that they are currently at. What amount are they going to increase to?
I ask because many publications print an answer that is 7.14% but I don't see that as correct because the first line would have to say an increase of 184 basis points for that to be correct. My answer to this is about 5.4%. Am I thinking correctly?
secondly, if rates are about to move to 7.14% from 5.3% what percentage move is this?
I get an answer of about 35%. am I off on this or is everyone else that I ask not calculating correctly? Answered by Claude Tardif. 





A 52 cubic foot box 
20060614 

From Konstanze: I need to figure out what LXHXW I need to create a 1.5 cubic meter or 52 cubic foot volumethere is an answer in your database that relates to this..but it does not give the formula to go from cubic feet/inches to a measurement for a box.
Empirically I can figure out that 3 x 3 x 3 equals 27 and that 3.5 feet (42") x 3.5 x 3.5 equals 42.87 and 3.75 x 3.75 x 3.75 equals 52.73 which is about 1.5 cubic meters (1cubic meter = 52.971643 cubic feet) i.e a box 45 x 45x 45 "
There must be an easier way. Answered by Penny Nom. 





A boat travels 60 miles upstream ... 
20060528 

From Teenarick: A boat travels 60 miles upstream against the current in 3 hours. It takes 2 hours for the return trip. Find the speed of the boat in still water. Answered by Penny Nom. 





integral of tan^4 x 
20060514 

From Aqil: integral of tan^{4} x Answered by Penny Nom. 





Small pipes and large pipes 
20060509 

From geece: A large fresh water reservoir has two types of drainage system. Small pipes and large pipes. 6 large pipes, on their own, can drain the reservoir in 12 hours. 3 large pipes and 9 small pipes, at the same time, can drain the reservoir in 8 hours.
How long will 5 small pipes, on their own, take to drain the reservoir? Answered by Penny Nom. 





Reverse percentage 
20060419 

From Mike: How would I calculate a reverse percentage?
Let me give you an example. I have two values that calculate to a success rate of 93.32. Total=1546051 Failures=103302. (1546051103302)/1546051=93.32% success.
Now how much would I need to increase the total value to get the success up to 95%? Answered by Stephen La Rocque. 





Interest computed on a 360 per year basis 
20060406 

From Sonya: Okay, I am reading this promissory note that says interest is computed on 365/360 basis by applying the ratio of annual interest rate (24%) over a year of 360 days.
I sure do not remember anything like this in school, and am stumped. Answered by Stephen La Rocque. 





Wes and Tony live 360 km apart. 
20060401 

From Thiru: Wes and Tony live 360 km apart. If Wes travels at 80 km/h towards Tony. And Tony travels at 100 km/h towards Wes, how long will it take before they meet. Answered by Penny Nom. 





Worker A can do a piece of work in 15 days,... 
20060323 

From Kenneth: Worker A can do a piece of work in 15 days, worker B in 12 days, and worker C in 10 days. Worker A works 2 days, worker B 3 days, and worker C 3 days. In what time can worker A and worker B finish the job by working together? Answered by Penny Nom. 





One worker can perform a certain job in 8 days 
20060228 

From Kenneth: One worker can perform a certain job in 8 days, another worker in 10 days and a third worker in 12 days. In what time can all three perform it working together? Answered by Stephen La Rocque. 





Related rates and an oil spill 
20060212 

From Brandon:
An Oil Tanker Spills 100,000 cubic meters of oil, which forms a slick that spreads on the water surface in a shape best modeled by a circular disc is increasing at a rate of 3m/min (it doesn't state what is increasing at 3m/min, so I'm assuming Radius until I can ask my teacher.) At t=T, the area of the "circular" slick reaches 100pi Sq. meters.
A) how fast is the area of the slick increasing at t=T
B)How fast is the thickness of the slick decreasing at t=T
C)Find the rate of change of the area of the slick with respect to the thickness at t=T.
Answered by Penny Nom. 





60.5 ft at 90 mi/hr takes how long? 
20060113 

From Bree: A ball travels over 60.5 ft at 90 mi/hr. How fast does the ball travel? Answered by Penny Nom. 





Two related rates problems 
20051229 

From Shimaera:
#1. A manufacturer determines that the cost of producing x of an item is C(x)=0.015x^{2}+12x+1000 and the price function is p(x)=250+2x/10. Find the actual and marginal profits when 500 items are produced.
#2. At 9 a.m a car is 10km directly east of Marytown and is traveling north at 100 km/h. At the same time, a truck leaves Marytown traveling east at 70 km/h. At 10 a.m, how is the distance between the car and the truck changing?
Answered by Penny Nom. 





One car leaves a spot traveling at 100 km per hour 
20051228 

From Jason: One car leaves a spot traveling at 100 km per hour. The second car leaves the same spot 15 minutes later and traveling at 120 km per hour. How long does it take for the second car to catch up to the first car? Answered by Penny Nom. 





Two dogs and a flea 
20051223 

From Michelle: Two dogs, each traveling at 10 ft/sec, run toward each other from 500 feet apart. As they run, a flea flies from the nose of one dog to the nose of the other at 25 ft/sec. The flea flies between the dogs in this manner until it is crush when the dogs collide. How far did the flea fly? Answered by Penny Nom. 





Pat invested a total of $3000 dollars 
20051214 

From Duane: Pat invested a total of $3000 dollars. part of the money yields 10 percent interest per year and the rest yields 8 percent interest per year if the total yearly interest is $256 how much did pat invest at 10 per cent and how much at 8 percent. Answered by Penny Nom. 





A loan of $50,000 
20051214 

From Fre: A loan of $50,000 taken today is payable within five years.
a. determine the annual payments within to be made to repay the loan if interest is charged at a rate of 10% compounded annually
b. show the amortization schedule Answered by Penny Nom. 





The bathtub curve 
20051013 

From David:
My father asked me to submit a question about the socalled 'bathtub
curve'. If you cut a bathtub in half lengthwise down it's middle, the
edge of the tub would describe the 'bathtub curve' which can be used
to demonstrate typical failure rates of products. This curve is
characterised by high initial (infant mortality) failure rates at
it's beginning, which drop quickly to a very low level. Failures then
increase gradually to the "end of life" stage where the failure rate
takes off dramatically again.
If anyone in the math department knows about the socalled 'bathtub
curve' my father would really appreciate the equation.
Answered by Chris Fisher and Edward Doolittle. 





A point is moving on the graph of x^3 + y^2 = 1 in such a way that 
20050917 

From Gina: A point is moving on the graph of x^{3} + y^{2} = 1 in such a way that its y coordinate is always increasing at a rate of 2 units per second. At which point(s) is the x coordinate increasing at a rate of 1 unit per second. Answered by Penny Nom. 





At what rate is the circumference of the circle increasing? 
20050808 

From John:
A mathematics professor is knitting a sweater. The main part of the sweater is knit in a large spiral, ending up with a diameter of 30 inches. She knits at a constant rate of 6/7 square inches per minute.
1. At what rate is the circumference of the circle increasing when the diameter is 2 inches?
2. How long will it take her to finish this piece of the sweater?
Answered by Penny Nom. 





A lighthouse is located on a small island,... 
20050714 

From Brittnee: A lighthouse is located on a small island, 3 km away from the nearest point P on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? Answered by Penny Nom. 





Three debts 
20050203 

From Kat: If I have three debts and 49% percent of total debt is loaned at 9% intrest, 34% of the debt is at 21% intrest and 17% of the total debt is at 14% intrest, how do I calculate the average rate of intrest on total debt? Answered by Penny Nom. 





Ratios and rates 
20041027 

From Kenneth: What is the difference between a ratio and a rate? Answered by Penny Nom. 





A rate of change problem 
20041015 

From Frank: Find the rate of change of the distance between the origin and a moving point on the graph of y = x(squared) + 1 if dx/dt = 2 centimeters per second. Answered by Penny Nom. 





$25,000 at 11% per year 
20041013 

From A student: suppose u have $25,000 to invest and the interest rate at your bank is 11%.
1) how much money would you have at the end of EACH of the first four years?
1) i already kno that the first year w/ interest is $27,750 but how do i get the 2nd? 3rd? 4th? years? Answered by Penny. 





Integrating e^sin(x) 
20040804 

From A student: I need to know that how to solve the integral " e^sin x", Answered by Harley Weston. 





Rate of change problems 
20040801 

From Jim: I just want to check a couple average rate of change problems because i just guessed on how to do them. Can you tell me how to do them?
the question says if f(x) = sqrt(x + 3), find f( x + rx). I got sqrt( x + rx + 3)
the other two are : f(x)= 3x1 (f(x)  f(1)) / (x1) ... I GOT 3 &
f(x)= x^3  x (f(x)  f(1)) / (x1) ... I GOT x^2 + x Answered by Penny Nom. 





Related rates and baseball 
20040426 

From Bethany: A baseball diamond is the shape of a square with sides 90 feet long. A player running from second to third base at a speed of 28 feet/ second is 30 feet from second base. At what rate is the player's distance from home plate changing? Answered by Penny Nom. 





Snow in the driveway 
20040409 

From Patricia: Following a severe snowstorm, Ken and Bettina Reeves must clear their driveway and sidewalk. Ken can clear the snow by himself in 4 hours, and Bettina can clear the snow by herself in 6 hours. After Bettina has been working for 3 hours, Ken is able to join her. How much longer will it take them working together to remove the rest of the snow? Answered by Penny Nom. 





A changing rectangle 
20040403 

From A student: The width x of a rectangle is decreasing at 3 cm/s,
and its length y is increasing at 5 cm/s. At what rate
is its area A changing when x=10 and y=15? Answered by Penny Nom. 





Some calculus problems 
20040401 

From Weisu:
I have questions about three word problems and one
regular problem, all dealing with derivatives.
 Find all points on xy=e^{xy} where the tangent line
is horizontal.
 The width x of a rectangle is decreasing at 3 cm/s,
and its length y is increasing at 5 cm/s. At what rate
is its area A changing when x=10 and y=15?
 A car and a truck leave the same intersection, the
truck heading north at 60 mph and the car heading west
at 55 mph. At what rate is the distance between the
car and the truck changing when the car and the truck
are 30 miles and 40 miles from the intersection,
respectively?
 The production P of a company satisfies the
equation P=x^{2} + 0.1xy + y^{2}, where x and y are
the inputs. At a certain period x=10 units and y=8
units. Estimate the change in y that should be made to
set up a decrease of 0.5 in the input x so that the
production remains the same.
If you could just give me some hints on these
questions, I'd really appreciate it. Thanks! Answered by Penny Nom. 





Distance, time and rate 
20040309 

From Peter: A car travels from one city to another at the speed of 32mph, if it had gone 4mph faster it could have made the trip in one half hour less. How far apart are the cities?
Answered by Penny Nom. 





A pyramidshaped tank 
20040213 

From Annette: The base of a pyramidshaped tank is a square with sides of length 9 feet, and the vertex of the pyramid is 12 feet above the base. The tank is filled to a depth of 4 feet, and water is flowing into the tank at a rate of 3 cubic feet per second. Find the rate of change of the depth of water in the tank. (Hint: the volume of a pyramid is V = 1/3 B h , where B is the base area and h is the height of the pyramid.) Answered by Harley Weston. 





Driving from Sarnia 
20040211 

From Jane: "Crystal drove from Sarnia at 80km/hr. Emily left Sarnia one hour later and drove along the same road at 100km/hr. How far from Sarnia did Emily overtake Crystal?" Answered by Claude Tardif. 





$20,000 at 25% compounded daily 
20040203 

From A student: Hi there, I'm tying to figure out how much interest I would pay on 20,000 if it was 25% compounded daily. Answered by Penny Nom. 





Business trip 
20031219 

From Ameer: A businnessman drives from Washington, D.C., to Boston, a distance
of 442 miles, and then makes the return trip. On the way to Boston,
he drives 65 miles per hour, taking an 1hour rest stop during the
drive. After finishing his business in Boston, he make the return
trip driving at 60 miles per hour and takes a 45minute rest stop
halfway through the trip. Which leg of the journey, Washington, D.C.
to Boston, or Boston to Washington, D.C., takes the longer time? Answered by Penny Nom. 





A helicopter rises vertically 
20030902 

From Kate: A helicopter rises vertically and t seconds after leaving hte ground its velocity is given in feet per second by v(t) = 8t + 40 / (t+2)^{2} How far above the ground will the helicopter be after 3 seconds? Answered by Penny Nom. 





Odd powers of sine and cosine 
20030625 

From Antonio: Can you please tell me how to integrate a trig function involving sine and cosine? I know if the powers of both the sine and cosine are even and nonnegative, then I can make repeated use of the powerreducing formulas. But for the question I have on my hand, the powers of both sine and cosine are odd: ( sin3x + cos7x ) dx. Answered by Harley Weston. 





Interest compounded daily 
20030624 

From Jeff: What would be the amount of interest charged or accrued and how is it calculated on principal balance of $209.12 @ 6% interest rate compounded daily for 70 days? Answered by Penny Nom. 





Percentage change in completion rate 
20030221 

From Remo: In the year 2000 300 out of a possible 600 completed a phone survey for a total of 50% completion rate. In the year 2001 20 out of 100 possible people completed the survey when it was redone. This is a completion rate of 20% To figure out the percentage change between the years can just use the difference between the percentage figures I already have? Or can I calculate it the same as you would for nonpercentage numbers. For example you gave an example here of a salary increase from $20 to $95 being an increase of 375%. Would I solve my problem the same way expect substituting my percentages (.5 and .2) in place of $20 and $95? Answered by Penny Nom. 





Storyteller figurines 
20030210 

From A student: It takes 3/4 of an hour to bake a storyteller figurine. If only one figurine can be baked at a time, how many can be baked in 6 hours? Answered by Penny Nom. 





Two airplanes leave Dallas 
20030206 

From A student: TWO AIRPLANES LEAVE DALLAS AT THE SAME TIME AND FLY IN OPPOSITE DIRECTIONS. ONE AIRPLANE TRAVELS 80 MILES PER HOUR FASTER THAN THE OTHER. AFTER THREE HOURS, THEY ARE 2940 MILES APART. WHAT IS THE RATE OF EACH AIRPLANE? Answered by Penny Nom. 





A train with 2 cars 
20030204 

From Michael: A train with 2 cars is traveling at a speed of 80 km/hr from town X to town y, located 800 km from each other. At the same moment that the train departed, a passenger started to walk back and forth from one end of car B to the other at a speed of 100cm/sec. Arriving in town Y, the passenger had already gone and returned 720 times. The length of car A is that of car B plus one fourth of the length of the locomotive, and the length of the locomotive equals the length of Car A plus one fifth of the length of car B. What is the total length of the train? Answered by Penny Nom. 





Integration of 1/(2+cos(x)) 
20030107 

From A student: integral from pi to 0 of dx/(2+cos x) i used the substitution t=tan(x/2) and i ended up with integral from +infinity to 0 of 2dt/(t^{2}+3) which looks like an inverse tan function , and i ended up with sqr(27)/2 pi , which is not the same as my calculator's answer , so i suspct i am doing some thing wrong. can some one tell me where i am going wrong please. Answered by Penny Nom. 





Miles per hour 
20021128 

From Liz: If a car has traveled 16 miles in 30 minutes, how many miles per hour did they go? Answered by Penny Nom. 





Filling A swimming pool 
20021121 

From Sarah: A swimming pool is being filled by three pumps. Alone pump A would take 6 hours, pump B would take 3 hours, and pump C would take 3 hours. If all three pumps are used to fill the pool, what fraction of the process is pump A. Answered by Penny Nom. 





Two tanks of water 
20021108 

From A student: A 2000 L tank containing 550 L of water is being filled with water at the rate of 75 L per minute from a full 1600 L tank. How long will it be before the two tanks have the same amount of water? Answered by Penny Nom. 





Two problems 
20021014 

From Seçkin: i am from turkey i am a teacher in a collage i have two difficult question which i havent solve yet these are very important for me.... Answered by Claude Tardif. 





Two rate problems 
20020930 

From Rebecca:
 There are two small holes in the bottom of a tub filled with water. If opened, one hole will empty the tub in three hours; the other will empty it in six hours. If both holes are opened at the same time, how long will it take to empty the tub?
 An airplane flies 400 miles per hour in calm air. It can cover 900 miles flying with the wind in the same time that it can cover 700 miles against the wind. What is the speed of the wind?
Answered by Penny Nom. 





Painting a car 
20020924 

From A student: Dan can paint a car in 4 hours. Luke can pain the same car in 6 hours. Working together, how long would it take them to pain the same car? Answered by Penny Nom. 





How long will it be before you reach your friend? 
20020916 

From Margaret: Your friend runs at 8 miles per hour and you drive at 40 miles per hour, hence the diatance between you is decreasing at 32 miles per hour. You firend left 2 hours before you so how far has he gone in that 2 hours? How long will it take for the distance between you to decrease to 0 miles if the distance is decreasing at 32 miles per hour? Answered by Penny Nom. 





The 20 kilometer walk 
20020820 

From A student: THE 1996 OLYMPIC GOLD METAL WINNER FOR THE 20 KILOMETER WALK WAS JEFERSON. HIS TIME WAS 1HOUR, 20 MIN,AND 7SECONDS. HIS TIME WAS NOT GOOD ENOUGH TO BEAT THE WINNER IN 1988 BY JOSEF. HIS TIME WAS 1 HOUR,19 MIN, 57 SECONDS. WHAT WAS THE WALKING RATE OF EACH PERSON? Answered by Penny Nom. 





Integrating x^x 
20020618 

From Jeremy: I am a student at the University of Kansas and I am wondering if there is a general antiderivative for x^{ x} (i.e. the integral of x^{ x} dx)? I've looked in a bunch of Table of Integrals and have found nothing (can you guys find it?), so I'm sort of wondering if this may be a research type question. Answered by Claude Tardif. 





A good rule of thumb when driving 
20020613 

From Lisa: A good rule of thumb when driving is that you should be about one car length away from the car in front of you for every 10 miles per hour that you are travelling. Suppose you follow this rule perfectly (so you are exactly the correct distance away). You are waiting at a stop light with your front bumper just touching the car in front of you. The light turns green and the car in front accelerates at a constant rate "r". Calculate how you should accelerate in order to follow the rule. Answered by Penny Nom. 





Conics 
20020529 

From Brooke: Which conic cannot be generated by an intersection of a plane and a double napped cone? Answered by Chris Fisher. 





A spotlight shines on a wall 
20020525 

From Barb: A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight toward the bldg at a speed of 1.6 m/s, how fast is his shadow on the bldg decreasing when he is 4m from the bldg? Answered by Penny Nom. 





Trains and cracker boxes 
20020429 

From Lindsay: 1)Two trains are 250 miles apart on the same track heading towards eaxh other at 7AM The first train travels at 55mph, the second at 75mph. At what time would they crash? 2)A cracker box measures 12 by 2 by 18 inches. The company wants to double the amount of crackers, but keep the shapes the same[similar] Find the new length[nearest tenth of an inch] Answered by Penny Nom. 





A test drive 
20020423 

From Jennie: At 9 A.M. a test car driving at a constant speed passes a marker 50 miles from it's starting point. At noon the car is about 130 miles from the marker. If the test drive ends at 1:30 P.M., how far will the car be from its starting point? Answered by Penny Nom. 





Arc length 
20020417 

From Vix: Find the point on the curve r(t)=(12sint)i(12cost)j+5tk at a distance 13pi units along the curve from the point (0,12,0) when t=0 in the direction opposite to the direction of increasing arc length. Answered by Harley Weston. 





Related rates 
20020417 

From Molly: A tanker spilled 30 ft cubed of chemicals into a river, causing a circular slick whose area is expanding while its thickness is decreasing. If the radius of the slick expands at the rate of 1 foot per hour, how fast is them thickness of the slick decreasing when the area is 100 feet squared? Answered by Penny Nom. 





The bleep test 
20020413 

From Lorraine: I am currently undergoing training and have asked about a fitness test called the bleep test. The bleep test involves running continuously between two points that are 20m apart. These runs are done in time to a prerecorded bleep sounds on a prerecorded audio cassette. The time between the recorded bleeps decrease after each minute. I would like to do my own personal version for training but I have to work out the following before doing this: I need to find out the time inbetween the beeps at least for the first level so I can work out the difference for the other levels: The information given to me for the first level is : A 20 meter run at 8.5km/hr (how long would it take?) Please let me know if there is any way in working out this query with the information given. Answered by Leeanne Boehm. 





Mr. Byrd's drive to work 
20020322 

From Mr. Bollen: Mr. Byrd leaves his house at 7:00am to go to work. If he drives 40 mph he will arrive 3 minutes late and if he drives 60mph he arrives 3 minutes early. At what speed will Mr. Byrd have to drive in order arrive exactly on time. Please describe how you arrived at your answer. Answered by Claude Tardif and Penny Nom. 





15 miles at 13 miles per hour 
20020312 

From Rich: If I were traveling 15 miles at 13 miles per hour how long will it take? Answered by Penny Nom. 





A man and his wife walk up a moving escalator 
20020213 

From Monty: A man and his wife walk up a moving escalator. The man walks twice as fast as his wife. When he arrives at the top, he has taken 28 steps. When she arrives at the top, she has taken 21 steps. How many steps are visible in the escalator at any one time. Answered by Peeny Nom and Claude Tardif. 





The famous "train problem" 
20020112 

From Karen: How do you solve the famous "train problem"? For example, 2 trains leave different stations at the same time. One is traveling at 50 mph and the other at 40 mph. How long does it take them to meet? Answered by Penny Nom. 





Making toys 
20011217 

From Karen: Jonathan and Morgan are two of Santa's elves. Jonathan can make a toy in eleven hours. Morgan can make the same toy in nine hours. How long it take for both Jonathan and Morgan to make the toy if they were working together? Answered by Claude Tardif. 





A lighthouse and related rates 
20011129 

From Melissa: A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline, and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? Answered by Penny Nom. 





Julia goes to work 
20010925 

From Karyna: On one October morning, Julia rides her bike (at 12 miles per hour) from her home to her friend Ida's house; then the two of them walk (at 6 miles per hour) to work. If it takes an hour for Julia to go the 10 miles from her home to work, how far does she walk? Answered by Penny Nom. 





Ray and Jane 
20010906 

From A student: If Ray and Jane live 150 miles apart and they both leave their house at the same time, Ray goes 30 mph the whole time and Jane goes 50 mph the whole time how many miles are they from Ray's house when they meet? And what is the exact travel time untl they meet. Answered by Penny Nom. 





Joe and his dad 
20010828 

From Sarah: Joe Spout left a campsite on a trip down the river in a canoe, traveling at 6 km/h. Four hours later, Joe's father set out after him in a motorboat. The motorboat traveled 30 km/h. How long after Joe`s father started did he overtake the canoe? How far had Joe traveled down the river when his father overtook him? Answered by Leeanne Boehm. 





How far does the fly fly? 
20010807 

From Harold:
6 MPH 4 MPH
Rachel  Eli
10 Miles apart
The fly is on Rachels handlebars. The fly is scared so it flys back and forth at 20 MP H. How far has the fly flown when Rachel and Eli meet? f Answered by Penny Nom. 





Investing $5,000 
20010709 

From A student: A principal amount of $5,000 was invested in a savings account for 5 years.The interest earned was $500.Use the simple interest formula to find the annual rate of interest. Answered by Penny Nom. 





Area between curves 
20010613 

From Phil:
question 1 find the area bound by the curves y = x^{2} + 2x + 3 and y = 2x + 4 question 2 Find the volume generated by rotating the curve x^{2} + y^{2} = 9 about the xaxis Answered by Harley Weston. 





National consumption function 
20010509 

From Brian: If consumption is $11 billion when disposable income is 0 and the marginal propensity to consume is dC/dy = 1/(2y+4)1/2+0.3(in billions of dollars), find the national consumption function. Answered by Harley Weston. 





Working together 
20010426 

From Stephanie: Tom takes 10 hours to piant a mural on the wall of Evergreen School. Carol takes 6 hours to do the same job. If they work together, how long will it take them to paint the mural? Answered by Claude Tardif and Penny Nom. 





Oil revenue 
20010421 

From Brian: Suppose that t months from now an oil well will be producing crude oil at the rate of r(t), not a constant, barrels per month and that the price of crude oil will be p(t), not a constant, dollars per barrel. Assume that the oil is sold as soon as it is extracted from the ground.  Find an expression for the total revenue from the oil well, R(t).
 A certain oil well that currently yields 400 barrels of crude oil a month will run dry in 2 years. The price of crude oil is currently $18 per barrel and is expected to rise at a constant rate of 3 cents per barrel per month. What will be the total revenue from this well? {Hint: Model the degraded production rate with the equation:
r(t) = (ABt)e^{0.04t}} Answered by Harley Weston. 





Two boats 
20010419 

From Pat: Two boats head directly toward each other, one of them traveling 12 miles per hour and the other traveling 17 miles per hour. They begin at a distance of 20 miles from each other. How far apart are they one minute before they collide? Answered by Penny Nom. 





Cash advance 
20010328 

From A student: dulani has a new credit card. it says: you can get cash advances wherever you are. whenever you want. also if you pay off your balance in full each month for a small transaction fee, the cash is interest free. (see important information on reverse side. On reverese side it says: cash advance transaction fee: $500 or less 2.5%; $500.01 to $1000.002%; $1000.01 or more 1.5%; $2.00 minumum.
Annual percentage rater for cash advances 19.8%. dulani wants to know what a cash advance will really cost. Analyze the cash advance terms given here. how much would he pay for a $20 cash advance? What about a $450 cash advance? What annual interest rate are these charges equivalent to? choose other amounts. determine cost and annual ratees for these as well. make recommendations to dulani. Answered by Penny Nom. 





Two ferry boats 
20010325 

From Gil: Two ferry boats leave from opposite shores. One is faster than the other. They meet 720 yards from the nearest shore. They proceed to destination and upon returning they meet 400 yards from the other shore. What is the exact width of the river. Answered by Penny Nom. 





Processing speed 
20010126 

From Zac: A COMPUTOR IS ADVERTISED AS HAVING A PROCESSING SPEED OF 11 MILLION INSTRUCTIONS PER SECOND. ON THE AVERAGE, HOW LONG DOSE IT TAKE TO PROCESS ONE INSTRUCTION AT SUCH A SPEED? Answered by Leeanne Boehm. 





How do you integrate secant(theta)? 
20001222 

From Robert Williamson: How do you integrate secant(theta)? I know the answer is ln [sec(theta) + tan(theta)] but how do you get there? Answered by Claude tardif. 





Comparing an integral and a sum 
20001121 

From Douglas Norberg: A fellow teacher asked me about a problem she wanted to give to her students. It involved whether to take a million dollars or a penny doubled a number of times. I was able to determine the number must have been .01 * 2^{30} which is about $10 million and a lot more than $1 million. To check that I was right I used a spreadsheet and did a Riemann sum. When I finished I reasoned that I had done the task in several steps and I could have done it in 1 step. Thus I integrated .01 * 2^{x} from 0 through 30 but the number I got was $15,490,820.0324. Why the difference? Answered by Harley Weston. 





Cutting the lawn 
20001026 

From Kacie: Ellen can mow a lawn in 2 hours. Mary can mow the same lawn in 1.5 hours. How long would it take them to mow the lawn together? Answered by Penny Nom. 





Working together on a job 
20001023 

From Nicole: WORKING TOGETHER ON A JOB: Patrice, by himself can paint 4 rooms in 10 hours. If he hires April to help they can do the same job together in 6 hours. If he lets April work alone , how long will it take her to paint 4 rooms? Answered by Claude Tardif. 





Isolating an exponent 
20000924 

From C. Scott: Suppose you invest $500.00 in an account that pays 10% interest compounded annually. How long will it take for this value to triple? A=p(1 + i)^{a} A=1500 i=0.1 1500=500(1.1)^{a} p=500 a=unknown How do you solve this problem algebraicly? How do you isolate the variable (a) when it is an exponent? Answered by Harley Weston. 





Velocity of a pendulum 
20000828 

From Mekca: A pendulum hangs from the ceiling. as the pendulum swings, its distance,d cm, form one wall of the room depends on the number of seconds,t, since it was set in motion. assume that the equation for d as a function of t is: d=80+30cos3.14/3t, t>0. estimate the instantaneous rate of change of d at t=5 by finding the average rates for t=5 to 5.1, t=5 to 5.01, and t=5 to 5.001. Answered by Harley Weston. 





A problem with a quadratic 
20000809 

From David Xiao: Find the value of a such that 4x^{2} + 4(a2)x  8a^{2} + 14a + 31 = 0 has real roots whose sum of squares is minimum. Answered by Harley Weston. 





Calculus Research Questions 
20000522 

From William Wright: I am a Calculus Teacher, and me and my class ran into these two problems without solutions in my manual, we got answers, but are unable to check them. If anyone gets this email and can respond to this with the solutions it be greatly appreciated. . . . Answered by Harley Weston. 





Related Rates 
20000507 

From Derek: How can you show that if the volume of a balloon is decreasing at a rate proportional to its surface area, the radius of the balloon is shrinking at a constant rate. Answered by Harley Weston. 





Two calculus problems 
20000501 

From Kaushal Shah: How Do WE Integrate the following Functions,  Integral xtanx dx
 How was natural base "e" discovered and why e=2.7.......
Answered by Claude Tardif. 





Compounding continuously 
20000321 

From Gina: You deposit $1500 in an account that pays 6.5% annual interest, compounded continuously. Find the balance after 10 years. I'm not sure what to do with the "compounded continuously" part. Answered by Penny Nom. 





Travelling from A to B 
20000320 

From Matt: A car is traveling from point A to point B. Point A and point B are 30 miles apart. A car travels 30 miles an hour from point A to point B and then goes back to point A. How fast does the car have to go back to average 60 miles an hour? Answered by Claude Tardif. 





Two calculus problems 
20000303 

From Tara Doucet:
The height of a cylinder with a radius of 4 cm is increasing at rate of 2 cm per minute. Find the rate of change of the volume of the cylinder with respect to time when the height is 10 cm. A 24 cm piece of string is cut in two pieces. One piece is used to form a circle and the other to form a square. How should the string be cut so the sum of the areas is a maximum? Answered by Harley Weston. 





Saving for college 
20000301 

From Andrew Kunz: SAVING FOR COLLEGE In this project, you will forecast a friend's finances. Jane has received $75 from her grandparents on every birthday since she was one year old. She has been saving the money in an account that pays 5% interest. She is saving her money to help you pay for her college education, which she will start this fall after her 18th birthday. She also has been receiving birthday checks from her other relatives, but these didn't start until she was 12 years old. The amounts of these checks from her 12th birthday until her 18th birthday are $45, $45, $55, $50, $55, $60, $65. How much money will she have saved just from her birthdays by the time she starts college? IS this a reasonable amount to pay for a used car during her junior year in college? If she had invested her money in a different accoutn that had earned 7% interest, how much more money would she have saved? Answered by Penny Nom. 





A moving point on the graph of y=sinx 
20000222 

From Veronica Patterson: Find the rate of change of the distance between the origin and a moving point on the graph of y=sinx if dx/dt=2 centimeters per second. Answered by Harley Weston. 





Jogging up a hill 
20000217 

From Aaron Williams: A man jogs down the hill at 6 miles per hour and back up at 4 miles per hour. The total time that he travels is 5/4 hours. What is the total distance that he jogged. The solution is supposedely 3 miles, but i believe that it is 6. Can you help me? Please. Answered by Claude Tardif. 





Play ball 
20000203 

From Jessie: Here's a calc question that is probably a lot easier than I am making it. If you have a legendary "baseball problem" for the related rates section of Calc I, and you are given that the runner is running from 2nd to 3rd base at a given rate, and the umpire is standing at home plate, and you are given the distance between the bases on the field, how do you find the rate of change of the angle between the third base line (from the point of the umpire) and the runner? Here is a sample prob: Runner is moving from 2nd to 3rd base at a rate of 24 feet per second. Distance between the bases is 90 feet. What is the rate of change for the angle (theta, as described previously) when the runner is 30 feet from 3rd base? Answered by Harley Weston. 





Capacitor discharge rate 
20000109 

From Bill Phillips: I need to be solve for t in the following rate problem for an electrical capacitor. Vr = E(e^t/RC), e=nat. log 2.718 raised to the t/RC power. Answered by Harley Weston. 





A decreasing ellipsoid 
19991215 

From A student instructor: The volume of an ellipsoid whose semiaxes are of the lengths a,b,and c is 4/3 *pi*abc. Suppose semiaxes a is changing at a rate of A cm/s , the semiaxes b is changing at B cm/s and the semiaxes c is changing at C cm/s . If the volume of the ellipsoid is decreasing when a=b=c what can you say about A,B,C? Justify. Answered by Harley Weston. 





Two calculus problems 
19991213 

From Alan: I have 2 questions that are very new to me, they were included on a quiz and the material was never covered. Our teacher never explained the purpose and detailed explanation of how to solve the problem. Could you help? Thanks. Question 1: A ball is falling 30 feet from a light that is 50 feet high. After 1 sec. How fast is the shadow of the ball moving towards the light post. Note that a ball moves according to the formula S=16t^2 Question 2: How many trapezoids must one use in order for the error to be less than 10^8 if we want to find the area under the curve Y=1/X from 1 to 2. Find the exact area, Graph the function and use the trap rule for the "N" that you found. Answered by Harley Weston.






Two calculus problems 
19991201 

From O'Sullivan: Question #1 Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely? It's under the chain rule section of differentiation if that any help. I've set up a ratio and tried to find the constant but am stuck. Question #2 The figure shows a lamp located three units to the right of the yaxis and a shadow created by the elliptical region x^2 + 4y^2 < or= 5. If the point (5,0) is on the edge of the shadow, how far above the x axis is the lamp located? The picture shows an x and y axis with only the points 5 and 3 written on the x axis. the lamp is on the upper right quadrant shining down diagonally to the left. There's an ellipse around the origin creating the shadow. It's formula is given as x^2 + 4y^2=5. Answered by Harley Weston. 





Clockwise or Counterclockwise? 
19991027 

From Tim: A particle moves around the circle x^{2} + y^{2} = 1 with an xvelocity component dx/dt = y  Find dy/dt
 Does the particle travel clockwise or counterclockwise around the circle? Why?
Answered by Harley Weston. 





Working Backwards 
19991016 

From Linda: I am having problems finding examples of problems that require "Working Backwards" used as a strategy for solving. We are required to give a presentation on Monday, October 25, 1999 in our school districts math class. We are trying to become better problem solvers and how to teach problem solving in the elementary classroom. Help! I can't find anything in my web searches. Answered by Penny Nom. 





Solving using logs 
19991011 

From Rich Bolton: Here's my question: $541.39(1+i)^{15}=784.09 Can you please show me how to do this? Answered by Penny Nom. 





A circle in a square 
19990526 

From Jose V Peris: A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle expands, the square expands to maintain the condition of tangency. find the rate at which the perimeter of the square is increasing. find the rate of increase in the area enclosed between the circle and the square at the instant when the area of the circle is 25(pi) square inches. Answered by Harley Weston. 





Related rates 
19990513 

From Tammy: The sides of a rectangle increase in such a way that dz/dt=1 and dx/dt=3*dy/dt. At the instant when x=4 and y=3, what is the value of dx/dt? (there is a picture of a rectangle with sides x and y, and they are connected by z, which cuts the rectangle in half) Answered by Harley Weston. 





A bike race 
19990423 

From Bill Gepford: Bill was in a bike race and his friend kyle calculated that if he went 15mph that he would cross the noontime checkpoint one hour early but if he rode 10mph he would arrive one hour late. How far away is the checkpoint? Answered by Penny Nom. 





The average rate of change of a function 
19990420 

From Tammy: Suppose that the average rate of change of a function f over the interval from x=3 to x=3+h is given by 5e^h4cos(2h). what is f'(3)? I would appreciate any help with this question. Answered by Harley Weston. 





An airplane problem 
19990308 

From B.M.R.: A plane left New York and headed East to its destination 3600 miles away across the Atlantic. On the way back its speed was boosted by a 50 mph tail wind and it arrived an hour early. What was its normal speed? Answered by Jack LeSage. 





Lunes 
19990204 

From Kai G. Gauer: A prof once told me that a certain type of lune is quadrable given that the diameter is an integer. She used the construction of a right isosceles triangle within a semicircle and later constructed another semicircle on the base of the first semicircle and used area subtraction to show equality to a smaller triangle with quadrable area. What happens when the original inscribed triangle is no longer isosceles? She mentioned something about other lunes also being quadrable; but not all. What are the dimensions of other such lunes? Note: I'm not certain if I still have my hercules account; please simply post on Q&Q. Thanks! Answered by Chris Fisher. 





Triangular Numbers 
19981030 

From Matt: i would like to know about triangular numbers and it history i would also like to know about the history of prime numbers thank you Answered by Chris Fisher. 





Problem Solving 
19981015 

From Pamela Fisher: Can you give me a comprehensive list of problem solving stategies. I teach grade one in a K to 8 school and we are working on improving problem solving at all grade levels. I have heard that there is a list of various strategies that we could adapt to different grade levels. Any help you give me would be appreciated. Thank you. Pamela Fisher Answered by Walter Whiteley. 





A Calculus Problem 
19980628 

From Lorraine: I'm a postsecondary student taking calculus by correspondence. I'm stuck on the following question (and similar ones) Can you help? Evaluate the following indefinite integral: d(theta)  1 + sin (theta) (It says to multiply both numerator and denominator by: 1  sin(theta) Thanks Lorraine Answered by Harley Weston. 





A Car Wash 
19980404 

From Lisa GotimerStrolla: Al washes a car in six hours. Fred washes a car in eight hours. How long will it take them to wash a car together? Answered by Harley Weston. 





A Tightrope Walker. 
19980219 

From Amy Zitron: A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A.... Answered by Harley Weston. 





Bathtub 
19980104 

From Jeffrey Yau: A bathtub, with two taps, can be filled in 20 minutes using only the cold water tap. It can be filled in 30 minutes using only the hot water tap. The flow of each tap is not changed when both taps are turned on. It takes 24 minutes to drain the full tub. Starting with an empty tub and the drain plug in place, the cold water turned on. Five minutes later the hot water is also turned on, and five minutes after that the drain plug is removed. How many additional minutes, after the plug is removed, would it take to fill the tub? Answered by Harley Weston. 





Ajax, Beverley, Canton and Dilltown 
19970314 

From S. Johnson: The following towns are placed on a coordinate system. Ajax at (x,z), Dilltown at (10,0), Canton at (0,0) and Beverly at (0,10). The roads from Beverly to Canton and from Canton to Dilltown are perpendiculat to each other and are each 10 miles in length. A car traveling at all times at a constant rate, would take 30 minutes to travel straight from Ajax to Canton, 35 minutes to travel from Ajax to Canton via Beverly, and 40 minutes to travel from Ajax to Canton via Dilltown. What is the constant rate of the car, to the nearest tenth of a mile per hour. Answered by Chris Fisher and Harley Weston. 

