







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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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 
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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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 
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. 





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. 





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. 





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. 





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 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. 

