AP Calculus



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Related Rates Practice Problems (3)

1. Sand falls from a conveyor belt at the rate of 10 cubic meters per minute onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the height and the radius changing when the pile is 4 meters high?

2. Water is flowing at the rate of 50 cubic meters per minute from a shallow concrete conical reservoir (vertex down) of base radius 45 meters and height 6 meters. How fast is the water level falling when the water is 5 meters deep? How fast is the radius of the water’s surface changing?

3. Water is poured into a conical cup at the rate of 2/3 cubic in/sec. If the cup is 6 inches tall and the top of the cup has radius 2 inches, how fast does the water level rise when the water is 4 inches deep?

4. A water trough is 12 feet long and its cross-section is an equilateral triangle with sides 2 feet long. Water is pumped into the trough at a rate of 3 cubic feet per minute. How fast is the water level rising when the depth of the water is ½ foot?

5. A conical tank with vertex down has a radius of 8 m at the top and is 12 m high. If the water flows into the tank at 10 m3/sec, how fast is the depth of the water changing when the height is 4 m?

6. A stone dropped into a still pond sends out circular ripples whose radius increases at a rate of 3 ft/sec. How rapidly is the area enclosed by the ripple increasing at the end of 10 sec?

7. A water trough is 10 ft long. The vertical ends are isosceles triangles 4 ft across the top with sloping sides of length 3 ft. Water is flowing into the trough at a rate of 2 cubic feet/min. Find the rate of change of the depth of the water in the trough when the depth is 1.5 feet.

8. Barney has a swimming pool in the shape of a triangular prism below. It’s a real conversation piece (. He is filling the pool at a rate of 3 cubic feet per second. How fast is the height of the water changing when the height is 6 ft?

9. Suppose the radius of a spherical balloon is shrinking at ½ cm per minute. How fast is the volume decreasing when the radius is 4 cm?

10. A spherical balloon is inflated at the rate of 3 cubic cm/min. How fast is the radius of the balloon changing when the radius is 6 cm?

SKIP! 11. Suppose a spherical balloon grows in such a way that after t seconds, the volume is given by [pic] cm3. How fast is the radius changing after 64 seconds?

12. A point moves around the circle x2 + y2 = 9. When the point is at [pic], its x-coordinate is increasing at the rate of 20 units per second. How fast is its y-coordinate changing at that instant?

13. A ladder 15 feet long leans against a vertical wall. Supposed that when the bottom of the ladder is x feet from the wall, the bottom is being pushed toward the wall at the rate of 1/2x feet/sec. How fast is the top of the ladder rising at the moment the bottom is 5 feet from the wall?

14. A board that is 5 feet long slides down a wall. At the instant the bottom end is 4 feet from the wall, the other end is moving at the rate of 2 ft/sec. At that moment,

a. how fast is the bottom end sliding along the ground?

b. how fast is the area of the region between the board, ground and wall changing?

15. An aircraft is climbing at a 30º angle to the horizontal. How fast is the aircraft gaining altitude if its speed is 500 mph?

16. A man 6 feet tall is walking at a rate of 3 ft/sec toward a streetlight that is 18 ft high. At what rate is his shadow length changing? How fast is the tip of his shadow moving?

17. Let A be the area of a square whose sides have length x and assume x varies with time. How are dA/dt and dx/dt related? At a certain instant, the sides are 3 ft long and growing at a rate of 2 ft/min. How fast is the area growing at that instant?

18. Let V be the volume of a cylinder having height h and radius r and assume h and r vary with time. At a certain instant, the height is 6 in and increasing at 1 in/sec, while the radius is 10 in and decreasing at 1 in/sec. How fast is the volume changing at that moment? Is the volume increasing or decreasing then?

19. Let ( (in radians) be an acute angle in a right triangle, and let x and y, respectively, be the lengths of the sides adjacent and opposite (. Suppose also that x and y vary with time. At a certain instant, x = 2 units and is increasing at 1 unit/sec, while y = 2 units and is decreasing at ¼ unit/sec. How fast is ( changing at that instant? Is it increasing or decreasing?

20. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mph. How fast is the radius of the spill increasing when the area is 9 square miles?

21. A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15 cm/min. At what rate must air be removed when the radius is 9 cm?

22. A 13-ft ladder is leaning against a wall. If the top of the ladder slips down that wall at a rate of 2 ft/sec, how fast will the foot be moving away from the wall when the top of 5 ft above the ground?

23. At a certain instant, each edge of a cube is 5 inches long and the volume is increasing at a rate of 2 in3/min. How fast is the surface area of the cube increasing?

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20 ft

9 ft

13 ft

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