Applications of Derivatives: Related Rates

Another application for derivatives is solving problems which relate the rate of change of one variable to the rate of change of another variable.

The rate of change of a variable is its derivative with respect to time. The rate of change of a distance, x is dx/dt.

Approach for solving a related rate problem:

The radius of a circle increases at a rate of 0.10inch per hour. What is the rate of increase of the area of the circle when the radius is 8 inches?

1. Define the problem. What rates are you trying to relate?

dr/dt and dA/dt, where r is the radius and A is the area.

2. Draw a sketch. Gather and summarize information. Remember the problem solving steps here.

dr/dt = 0.10 in/hr

3. Look for an equation that relates your two variables, r and A.

A = 3.14 r^{2}

4. Take the derivative of this equation with respect to time.

dA/dt = 3.14 * 2r dr/d

5. Now substitute in known values and solve the problem.

dA/dt = 3.14 * 2(8in) (0.1in/hr) = 5.024 in^{2}/hr

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