[Note for Mac users]
Up: Notes for Math 10

Notes on Related Rates:

Question: Suppose air is blown into a spherical balloon at a rate of 36 in3/sec. How fast is the radius of the balloon increasing at the instant that the radius is 3 inches?

A procedure for handling a related rates problem like the one above is as follows:

  1. Draw a picture of the situation.
    [balloon picture]

  2. Introduce variables for quantities that are changing. For example:
    Let V be volume and r the radius of the balloon.

    Warning: do not put a number on any quantity that is changing!

  3. Write down explicitly what rates are involved. For example:
    We know that dV/dt = 36 in3/sec.
    We want dr/dt when r = 3 in.

    Note: decreasing quantity means a negative derivative.

  4. Find an equation relating the quantities involved. For example:
    V = (4/3) p r3.

    These may come from:

  5. Differentiate both sides with respect to t (like implicit differentiation), considering the changing quantities as functions of t. For example:
    V = (4/3) p r3, so
    dV/dt = (4/3) p (3 r2 dr/dt) = 4 p r2 dr/dt

    Warning: don't forget the chain rule!

  6. Substitute the given rates and other constants. For example:
    36 = 4 p 32 dr/dt
    Warning: this comes after differentiating!

  7. Solve for the desired rate. For example:
    dr/dt = 36 / (4 p 32) = (36/36p) = 1/p.
    So dr/dt is approximately 0.32 in/sec.

Up: Notes for Math 10

Comments to: dpvc@union.edu
Created: Oct 14 1997 --- Last modified: Mar 1, 1999 12:07:48 PM