Exercises - Related Rates

  1. A stone dropped into a pond causes a series of concentric ripples in the form of circles. If the radius of the outer ripple increases steadily at the rate of 6 ft/sec, find the rate at which the area of the disturbed water is increasing when the radius is 8 feet.

  2. A 3 foot tall child runs away from a street light that is 13 feet high. How fast is the far end of his shadow moving given that the child is running at the rate of 2 feet per second? Also, how fast is the length of his shadow changing?

  3. Air is being pumped into a spherical balloon at the rate of $8\pi$ in$^3$/min. Find the rate of change of the radius when the surface area is $16\pi$ in$^2$.

  4. Water runs into a conical tank at the rate of $5\pi$ ft$^3$/min. The tank stands vertex down and has a height of 10 feet and a base radius of 5 feet. How fast is the water level rising when the water is 4 feet deep?

  5. The length of a rectangle is increasing at the rate of 3 in/min, while the width is decreasing at the rate of 2 in/min. Find the rate of change of the area of the rectangle when the length is 5 inches and the width is 3 inches.

  6. A boat is pulled in by means of a winch on a dock 12 ft above the deck of a boat. If the winch pulls in rope at the rate of 4 ft/sec, determine the speed of the boat when 13 ft of rope is out.

  7. The radius of a right circular cylinder is increasing at the rate of 3 cm/sec. If the volume of the cylinder remains constant, find the rate at which the height of the cylinder is changing when the radius is 5 cm and the height is 4 cm.

  8. A conical tank (vertex down) is 10 ft across the top and 12 ft deep. If water is flowing into the tank at the rate of 10 ft$^3$/min, find the rate of change of the depth of water the instant it is 8 feet deep.

  9. A weather balloon is released at 9:00 a.m. and rises vertically at the rate of 25 ft/min. An observer stands on level ground 400 feet from the balloon's release point. How fast is the distance between the balloon and the observer's feet changing at 9:12 a.m.?

  10. Water is flowing into a cylindrical tank of radius 2 ft at the rate of 8 ft$^3$/min. How fast is the water level rising?

  11. The volume of a right circular cone is increasing at the rate of $25\pi$ in$^3$ per minute and the diameter of the base of the cone is decreasing at the rate of 1 inch per minute. At a certain instant the base diameter is 12 inches long and the height is 18 inches. What is the rate of change in the height of the cone at this instant?

  12. The side of an equilateral triangle is $2\sqrt{3}$ inches long and is increasing at the rate of 4 inches per second. Find the rate of change in the area of the triangle at this moment.

  13. Water is poured into a cone at the rate of 12 ft$^3$ per minute. If the cone is 12 feet high and 6 feet in diameter, how fast is the water level rising when the water is 7 feet deep? (The vertex of the cone is down.)

  14. A spherical balloon is expanding under the influence of solar radiation. If its radius is increasing at the rate of 2 in/min, how fast is the volume increasing when the radius is 5 inches?

  15. A streat urchin is sitting at the base of a wall 5 feet high. He is holding one end of a string; a wharf rat is on the other end of the string. As the rat runs along the top of the wall, the urchin lets out string at the rate of 2 feet per second, but the string remains taut. Find the rate at which the rat is moving along the wall when 13 feet of string has been let out by the urchin. How does your answer change if the rat is running along the base of the wall?

  16. The base of a triangle is increasing at the rate of 3 inches per minute while the altitude is decreasing at the same rate. At what rate is the area changing when the base is 10 inches wide and the altitude is 6 inches high?

  17. A spotlight is on the ground 100 feet from a building that has vertical sides. A person 6 feet tall starts at the spotlight and walks toward the building at a rate of 5 feet per second. How fast is the top of the shadow moving \underline{down} the building when the person is 50 feet away from the building?

  18. The volume of a cylinder is increasing at the rate of $48\pi$ in$^3$ per second. The height of the cylinder is always twice the radius. Find the rate of change in the surface area of the cylinder when the volume of the cylinder is $128\pi$ in$^3$.

  19. A conical paper cup (vertex down) is leaking water at the rate of $4\pi$ in$^3$ per minute. If the cup is 12 inches high and 6 inches in diameter, at what rate is the water level being lowered at the instant the top surface area of the water is $4\pi$?

  20. Water is flowing out of a conical tank (vertex down) of height 10 feet and radius 6 feet, in such a way that the water level is falling 1/2 foot per minute. How fast is the volume of water in the tank decreasing when the water in the tank is 5 feet deep?

  21. A pole 10 feet long rests against a vertical wall forming a right triangle with the ground. Let $\theta$ be the angle between the top of the pole and the wall. If the bottom of the pole slides away from the wall, how fast does the area of the triangle change with respect to $\theta$ when $\theta = \pi / 6$?