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. 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?

  6. 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?

  7. 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 $\theta = \pi / 6$?