# Exercise on related rates and implicit differentiation

1. Each edge of a cube is expanding at the rate of 1 centimeter (cm) per second. How fast is the volume changing when the length of each edge is (a) 5 cm? (b) 10 cm? (c) cm?

2. An airplane flies in level flight at constant velocity, eight miles above the ground. (In this exercise assume the earth is flat.) The flight path passes directly over a point P on the ground. The distance from the plane to P is decreasing at the rate of 4 miles per minute at the instant when this distance is 10 miles. Compute the velocity of the plane in miles per hour.

3. A baseball diamond is a 90-foot square. A ball is batted along the third-base line at a constant speed of 100 feet per second. How fast is its distance from first base changing when (a) it is halfway to third base? (b) it reaches third base?

4. A boat sails parallel to a straight beach at a constant speed of miles per hour, staying miles offshore. How fast is it approaching a lighthouse on the shoreline at the instant it is exactly miles from the lighthouse ?

5. A reservoir has the shape of a right-circular cone. The altitude is 10 feet, and the radius of the base is 4 ft. Water is poured into the reservoir at a constant rate of 5 cubic feet per minute. How fast is the water level rising when the depth of the water is 5 feet if (a) the vertex of the cone is up? (b) the vertex of the cone is down?

6. A water tank has the shape of a right-circular cone with its vertex down. Its altitude is 10 feet and the radius of the base is 15 feet. Water leaks out of the bottom at a constant rate of 1 cubic foot per second. Water is poured into the tank at a constant rate of cubic feet per second. Compute so that the water level will be rising at the rate of 4 feet per second at the instant when the water is 2 feet deep.

7. Water flows into a hemispherical tank of radius feet (flat side up). At any instant, let denote the depth of the water, measured from the bottom, the radius of the surface of the water, and the volume of the water in the tank. Compute at the instant when feet. If the water flows in at a constant rate of cubic feet per second, compute , the rate at which is changing, at the instant when feet.

8. A variable right triangle in the -plane has its right angle at vertex , a fixed vertex at the origin, and the third vertex restricted to lie on the parabola The point starts at the point at time and moves upward along the -axis at a constant velocity of cm/sec. How fast is the area of the triangle increasing when sec?

9. The radius of a right-circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as the radius. When the radius is 1 foot the altitude is 6 feet. When the radius is 6 feet, the volume is increasing at a rate of 1 cubic foot per second. When the radius is 36 feet, the volume is increasing at a rate of cubic feet per second, where n is an integer. Compute .

10. A particle is constrained to move along a parabola whose equation is . (a) At what point on the curve are the abscissa and the ordinate changing at the same rate? (b) Find this rate if the motion is such that at time we have and .

11. The equation defines as one or more functions of . (a) Assuming the derivative exists, and without attempting to solve for , show that satisfies the equation . (b) Assuming the second derivative exists, show that whenever .

12. If , the equation defines as a function of . Without solving for , show that the derivative has a fixed sign. (You may assume the existence of .)

13. The equation defines implicitly as two functions of if . Assuming the second derivative exists, show that it satisfies the equation .

14. The equation defines implicitly as a function of . Assuming the derivative exists, show that it satisfies the equation

15. If , where is a rational number, say , then . Assuming the existence of the derivative , derive the formula using implicit differentiation and the corresponding formula for integer exponents.

# Exercise on Related Rates

1. The sides of the rectangle increase in such a way that and At the instant when and , what is the value of

2. A person 2 meters tall walks directly away from a street light that is 8 meters above the ground. If the person's shadow is lengthening at the rate of 4/9 meters per second, at what rate in meters per second is the person walking?

3. An observer at point A is watching balloon B as it rises from point C. The balloon is rising at a constant rate of 3 meters per second (this means dy/dt = 3) and the observer is 100 meters from point C.

(a) Find the rate of change in z at the instant when y = 50. (They want dz/dt.)

(b) Find the rate of change in the area of right triangle BCA when y = 50.

(c) Find the rate of change in when y = 50. (They want )

4. (Calculus classic) The bottom of a 10-foot ladder is going away from the wall at dx/dt = 2 feet per second. How fast is the top going down the wall? Draw the right triangle to find dy/dt when the height y is (a) 6 feet (b) 5 feet (c) zero.

5. The top of the 10-foot ladder can go faster than light. At what height y does dy/dt = c?

6. How fast does the level of a Coke go down if you drink a cubic inch a second? The cup is a cylinder of radius 2 inches first write down the volume.

7. A jet flies at 8 miles up and 560 miles per hour. How fast is it approaching you when (a) it is 16 miles from you; (b) its shadow is 8 miles from you (the sun is overhead); (c) the plane is 8 miles from you (exactly above)?

8. Starting from a 3-4-5 right triangle, the short sides increase by 2 meters/second but the angle between them decreases by 1 radian/second. How fast does the area increase or decrease?

9. A pass receiver is at x = 4, y = 8t. The ball thrown at t = 3 is at x = c(t - 3), y = 10c(t - 3).

(a) Choose c so the ball meets the receiver.

(b) At that instant the distance D between them is changing at what rate?

10. A thief is 10 meters away (8 meters ahead of you, across a street 6 meters wide). The thief runs on that side at 7 meters/ second, you run at 9 meters/second. How fast are you approaching if(a) you follow on your side; (b) you run toward the thief; (c) you run away on your side?

11. A spherical raindrop evaporates at a rate equal to twice its surface area. Find dr/dr.

12. Starting from P= V= 5 and maintaining PV= T, find dV/dt if dP/dt = 2 and dT/dt = 3.

13. (a)The crankshaft AB turns twice a second so d/dt =_____.

(b) Differentiate the cosine law 6 = 3 + x - 2 (3x cos ) to find the piston speed dx/dt when and

14. A camera turns at C to follow a rocket at R.

(a) Relate dz/dt to dy/dt when y = 10. (b) Relate d/dr to dy/dt based on y = 10 tan . (c) Relate d/dt to dy/dt and dy/dt.

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