1. Related Rates These problems find the rates of change of two or more related variables that are changing with respect to time, t. To begin, let’s examine the Volume of a balloon as it is filled with air.

2. The Volume of a sphere is given by: As a balloon is being filled, both the volume, V, and radius, r, increase with time, t. To examine the “rates of change” we need the derivative with respect to time, t.

3. For the derivative we want: Notice, the variables do not agree. The derivative is with respect to time so the Chain Rule must be followed.

4. On the left we get: On the right: Chain Rule

5. This gives the “Related rate”: Question: At a given time, what does the rate of change of the Volume depend upon? Answer: The radius, r, and its rate of change, dr/dt

6. Find how fast the volume of a balloon is changing when the radius is 6 inches and the diameter of the balloon is increasing at a rate of 3 inches per second. From the problem: r = 6 in and dr/dt = 1.5 in/sec

7. = 678.6What are the units? Look at the units in the equation: (inches) 2 (inches/sec) = in 3 /sec Which gives Volume/time