Related Rates Practice Worksheet

Related Rates Practice Worksheet

Read each problem carefully and show all your work. Remember to identify variables, write down known rates, formulate an equation relating the variables, differentiate implicitly with respect to time, and solve for the unknown rate.

1. A ladder 10 meters long is leaning against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 m/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 meters from the wall?

2. The radius of a circle is increasing at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 5 cm?

3. Sand is pouring from a chute at the rate of 10 cubic feet per minute. It forms a conical pile whose height is always equal to the radius of its base. How fast is the height of the pile increasing when the pile is 8 feet high? (Volume of a cone V = (1/3)πr²h)

4. In related rates problems, we differentiate with respect to a common variable, typically time (t).

5. A spherical balloon is being inflated. If its volume is increasing at a rate of 20 cubic cm per second, at what rate is its radius increasing when the radius is 3 cm? (Volume of a sphere V = (4/3)πr³)

6. When solving a related rates problem, the first step is to identify all given   and the rate that needs to be found.

7. After implicitly differentiating an equation relating variables, you will typically need to substitute known values to solve for the   rate.

8. A man 1.8 meters tall walks away from a lamppost 4.5 meters high at a speed of 1.2 m/s. How fast is the tip of his shadow moving when he is 10 meters from the lamppost?