Rough Suggested Algorithm for Related Rates Problems

1. Assign variables to all varying quantities.
2. Write both the given and the desired information in terms of these variables. Be sure to note which hold for all times and which hold only for an instant -- the latter can only be used after differentiation.
3. Find an equation relating the variables, and differentiate it with respect to time.
4. Substitute the given information and solve for the desired information.

Examples of Related Rates Problems

1. Sand runs onto a conical pile at the rate of 10 ft³/min. The pile's height is always equal to its base radius. How fast is its height increasing when it is 5 ft high?

2. A boat is pulled toward a dock by a rope from the bow to a ring on the dock 4 ft higher than the bow. If the rope is pulled in at 2 ft/sec, how fast is the boat approaching the dock when 10 ft of rope are out?

3. A balloon rising 15 ft/sec is 200 ft high when a car traveling 66 ft/sec passes directly beneath it. How fast is the distance between car and balloon increasing 1 sec later?

4. A police car is parked at a curb 6 m from the front of a row house. Its red light turns .5 rev/sec. A window in the house is 8 m from the closest point on the row house to the car. How fast is the spot of red light traveling as it passes the window?

5. A point moves along the curve x²-y²=9. When it passes the point (5,-4), its x-coordinate is increasing at the rate of 8 units/sec. How fast is its y-coordinate changing at that moment?

6. A piece of modeling clay has a volume of 4 in³, and remains in the shape of a cylinder. It is squeezed between flat surfaces so that its altitude is 1/2 in and decreasing 1/20 in/min. How fast is its radius changing?