3.9 Related Rates

In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity. The procedure is to find an equation that relates the 2 quantities and then use the Chain Rule to differentiate both sides with respect to time.

The key thing to remember is that the rates of change are derivatives.

Strategy to Solving Related Rates problems:

1) Read the problem carefully

2) Draw a diagram if possible

3) Introduce notation. Assign symbols to all quantities that are functions of time.

4) Express the given information and the required rates in terms of derivatives.

5) Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as will be shown in Example 3).

6) Use the Chain Rule to differentiate both sides of the equation with respect to t.

7) Substitute the given information into the resulting equation and solve for the unknown rate.

Example 1: A ladder (25 feet long) is leaning against a wall of a house. The base of the ladder is being pulled away from the wall at a rate of 2 feet per second.

a) How fast is the top of the ladder moving down the wall when the base of the ladder is 15 feet from the wall?

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b) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

Example #2: The volume of a tank is V =r³ , r is the radius of a tank. If the water is flowing into the tank at a rate of 15 ft³ /sec, find the rate at which the radius is changing when the radius is 3 feet.

V =r³ V =r³

r = 3 feet