Topic: rates list of topics
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	Rates & Ratios		2021-04-11
		From Kenneth: Is it correct to think that a rate can have units that are the same? Usually rates have units that are different to each other. Answered by Penny Nom.
	Related rates		2018-02-11
		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.
	Walking up and down a hill		2017-09-19
		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		2017-09-17
		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.
	Filling a pool with two hoses		2017-02-20
		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.
	Water leaking from a trough		2016-12-28
		From Kathryn: A trough is 6 m long, and has uniform cross-section 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.
	Two tax rates		2016-04-21
		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.
	Filling a tub		2015-01-02
		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 waste-pipe is closed. How much longer will it take to fill the bath? Answered by Penny Nom.
	Rates, percentages and units		2014-12-30
		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.
	Water in a conical funnel		2014-02-11
		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.
	Related rates		2014-01-30
		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		2014-01-29
		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.
	Water flowing out of a tank		2013-11-03
		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 cross-sectional 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.
	Related rates		2013-02-17
		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^2-1/3 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.
	Two cars approach a right-angled intersection		2012-04-10
		From Michael: Two cars approach a right-angled 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.
	Water is flowing into a cup		2011-12-19
		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.
	Two trains passing each other		2011-12-07
		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.
	Four carpenters can build eight houses in 10 days.		2011-11-23
		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		2011-11-21
		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.
	A reservoir has the shape of an inverted cone		2011-10-03
		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		2011-09-27
		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.
	How many meters long should the race be?		2011-05-26
		From Lee: Nathan can walk at a rate of two meters per second while David can easily go three-and-a-half 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		2011-04-17
		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		2011-04-06
		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.
	A camera's line of sight		2011-02-26
		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?		2011-02-26
		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		2011-01-30
		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.
	A circular oil slick of uniform thickness		2010-05-22
		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		2010-05-04
		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.
	Related rates and a rectangular sponge		2010-04-06
		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		2010-04-02
		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		2010-03-20
		From Luke: Answered by Harley Weston.
	A related rates problem		2010-03-03
		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		2010-02-20
		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.
	Related Rates Problem		2010-01-12
		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.
	Related Rates of a Cylinderical Trough with a Horizontal Axis		2009-12-26
		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		2009-12-16
		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.
	How fast is the distance between the two cars decreasing?		2009-12-08
		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.
	Mowing 17 lawns		2009-06-18
		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.
	Mixing drinks		2009-05-02
		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.
	Filling a container with water		2009-04-22
		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.
	Sand falls from a conveyor belt		2009-04-01
		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		2009-04-01
		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.
	Related rates		2009-03-14
		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		2009-03-11
		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		2009-03-09
		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.
	Three people working in pairs		2009-02-21
		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.
	Rates		2009-02-10
		From Jennifer: write each phrase in simplest form 30oz in 24 gl 48leaves on 9 plants Answered by Harley Weston.
	Water flowing from a cone to a cylinder		2009-01-23
		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.
	Related rates		2008-11-26
		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?		2008-11-22
		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		2008-11-12
		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		2008-11-10
		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.
	Water is leaking from a conical tank		2008-10-24
		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		2008-10-20
		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.
	Related rates		2008-10-16
		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		2008-10-15
		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.
	Liquid is being pored into the top of a funnel		2008-05-25
		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.
	Working together		2008-04-26
		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		2008-04-25
		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.
	A train and a boat		2008-03-15
		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.
	It takes Mark 3 hours to paint a picture		2008-02-27
		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.
	Two job offers		2008-02-10
		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.
	A spherical bubble gum bubble		2007-12-31
		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.
	Related rates - tree growth		2007-12-09
		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.
	Related Rates (streetlamp and shadow)		2007-11-09
		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.
	Related Rates (a water trough)		2007-11-07
		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.
	How to solve related rates problems		2007-10-27
		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		2007-10-26
		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		2007-10-22
		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		2007-10-18
		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		2007-10-18
		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		2007-10-15
		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.
	Water flowing into a tank		2007-09-21
		From andrew: Hi, I've been having real trouble visualizing this problem as apposed to a conical tank. It says the base of a pyramid-shaped 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.
	Water in a conical tank		2007-09-10
		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.
	A circular blob of molasses		2007-05-28
		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.
	A growing heap of sand: related rates		2007-04-23
		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		2007-04-19
		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.
	Water is being pumped into a trough		2007-04-09
		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.
	At what rate is the area of the triangle changing?		2007-02-24
		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.
	Water in a triangular trough		2007-01-30
		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.
	Wheat is poured on a conical pile		2006-11-17
		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.
	A melting snowball		2006-11-06
		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.
	Water is being pumped into the pool		2006-10-24
		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		2006-08-12
		From Erin: Water runs into a conical tank at the rate of 9ft3/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 r2 h). Answered by Penny Nom.
	If all of them work together, ...		2006-07-27
		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.
	Rates and percentages		2006-06-20
		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.
	Wes and Tony live 360 km apart.		2006-04-01
		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,...		2006-03-23
		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		2006-02-28
		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		2006-02-12
		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.
	Two related rates problems		2005-12-29
		From Shimaera: #1. A manufacturer determines that the cost of producing x of an item is C(x)=0.015x2+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		2005-12-28
		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.
	The bathtub curve		2005-10-13
		From David: My father asked me to submit a question about the so-called '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 so-called '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		2005-09-17
		From Gina: A point is moving on the graph of x3 + y2 = 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?		2005-08-08
		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,...		2005-07-14
		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		2005-02-03
		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		2004-10-27
		From Kenneth: What is the difference between a ratio and a rate? Answered by Penny Nom.
	Related rates and baseball		2004-04-26
		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		2004-04-09
		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		2004-04-03
		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		2004-04-01
		From Weisu: I have questions about three word problems and one regular problem, all dealing with derivatives. Find all points on xy=exy 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=x2 + 0.1xy + y2, 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.
	A pyramid-shaped tank		2004-02-13
		From Annette: The base of a pyramid-shaped 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.
	Storyteller figurines		2003-02-10
		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		2003-02-06
		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.
	Filling A swimming pool		2002-11-21
		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		2002-11-08
		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 rate problems		2002-09-30
		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		2002-09-24
		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.
	A good rule of thumb when driving		2002-06-13
		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.
	Related rates		2002-04-17
		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.
	A lighthouse and related rates		2001-11-29
		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.
	Working together		2001-04-26
		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.
	Two boats		2001-04-19
		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.
	Two ferry boats		2001-03-25
		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		2001-01-26
		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.
	Working together on a job		2000-10-23
		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.
	Related Rates		2000-05-07
		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		2000-03-03
		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.
	A moving point on the graph of y=sinx		2000-02-22
		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.
	Play ball		2000-02-03
		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.
	A decreasing ellipsoid		1999-12-15
		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		1999-12-13
		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		1999-12-01
		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 y-axis 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?		1999-10-27
		From Tim: A particle moves around the circle x2 + y2 = 1 with an x-velocity component dx/dt = y Find dy/dt Does the particle travel clockwise or counterclockwise around the circle? Why? Answered by Harley Weston.
	A circle in a square		1999-05-26
		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		1999-05-13
		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.
	Lunes		1999-02-04
		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.
	A Car Wash		1998-04-04
		From Lisa Gotimer-Strolla: 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.		1998-02-19
		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		1998-01-04
		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		1997-03-14
		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.

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