Name _________________________ Date ___________ Period



Name _________________________ Date ___________ Period ____________

Calculus Related Rates Worksheet

1. The radius r of a sphere is increasing at the uniform rate of 0.3 inches per second. At the instant when the surface area S becomes 100π square inches, what is the rate of increase, in cubic inches per second, in the volume V? [pic][pic]

(A) 10[pic] (B) 12[pic] (C) 22.5[pic] (D) 25[pic] (E) 30[pic]

2. The volume of a cone of radius r and height h is given by [pic]. If the radius and the height both increase at a constant rate of [pic] centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters?

(A) [pic] (B) 10[pic] (C) 24[pic] (D) 54[pic] (E) 108[pic]

3. The sides of the rectangle on the right increase in such a way that [pic] = 1 and [pic]. At the instant when x = 4 and y = 3, what is the value of [pic]?

(A) [pic] (B) 1 (C) 2 (D) [pic] (E) 5

4. A vertical circular cylinder has radius r feet and height h feet. If the height and radius both increase at a constant rate of 2 feet/sec, then the rate in ft2/sec at which the lateral surface area increases is:

(A) 4πr (B) 2π(r + h) (C) 4π(r + h) (D) 4πhr [pic] (E) 4πh

5. Two cars are traveling along perpendicular roads, car A at 40mph, and car B at 60mph. At noon, when car A reaches the intersection, car B is 90 mi away, and moving toward it. At 1 P.M. the rate, in miles per hour, at which the distance between the cars is changing is

(A) -40 (B) 68 (C) 4 (D) -4 (E) 40

6. A ladder 25 feet long is leaning against a house. The base of the ladder is pulled away from the house at a rate of 2 feet per second. How fast is the top moving down the wall when the base of the ladder is

a) 7 feet from the wall

b) 15 feet from the wall

c) 24 feet from the wall

7. A baseball diamond has the shape of a square with sides 90 feet long. A player 30 feet from third base (running toward third base) is running at a speed of 28 feet per second. At what rate is the player’s distance from home plate changing?

8. For the baseball diamond in exercise 10, suppose the player is running from first to second at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second.

9. A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light

a) at what rate is the tip of his shadow moving?

b) at what rate is the length of his shadow changing?

10. An airplane is flying at an altitude of 6 miles and passes directly over a radar antenna. When the plane is 10 miles away (s = 10), the radar detects that the distance s (from the radar antenna to the plane) is changing at a rate of 240 miles per hour. What is the speed of the plane?

11. The formula for the volume of a cone is[pic]. Find the rate of change of the volume if [pic] is 2 inches per minute and [pic]when

a) r = 6 inches

b) r = 24 inches

12. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when it is 15 feet high?

13. A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at the rate of 10 cubic feet per minute, find the rate of change of the depth of the water the instant it is 8 feet deep?

14. A trough is 12 feet long and 3 feet across the top. Its ends are isosceles triangles with an altitude of 3 feet. If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when it is 1 foot deep?

15. Find the indicated values of [pic] and[pic].

|Equation |Find |Given |Find |Given |

|I. [pic] |a) [pic] | x = 4 [pic]= 3 |b) [pic] |x = 25 [pic]= 2 |

|II. [pic] |a) [pic] |x = 3 [pic]= 2 |b) [pic] |x = 1 [pic]= 5 |

|III. [pic] |a) [pic] |x = 8 [pic]= 10 |b) [pic] |x = 1 [pic]= -6 |

|IV. [pic] |a) [pic] |x = 3 , y = 4 and[pic]= 8 |b) [pic] |x = 4, y = 3 and[pic]= -2 |

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