Up: Notes for Math 13

Notes on Related Rates:

A procedure for handling related rates problems is as follows:
  1. Draw a picture of the situation.

  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:
    Given: dV/dt = 8 in3/min
    Want: dr/dt when r = 2 in.

    Note: decreasing quantity means a negative derivative.

  4. Find an equation relating the quantities involved. For example:
    V = (4/3)pi 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) pi r3, so
    dV/dt = (4/3) pi (3 r2 dr/dt) = 4pi r2 dr/dt

    Warning: don't forget the chain rule!

  6. Solve for the rate you want. For example:
    dr/dt = (dV/dt) / (4pi r2)

  7. Evaluate using the given rates and other values. For example:
    dr/dt = 8 / (4pi 22) = (8/16pi) = 1/(2pi)
    So dr/dt is approximately 0.16 in/min.

Up: Notes for Math 13

Comments to: dpvc@union.edu
Created: Oct 14 1997 --- Last modified: Oct 27, 1998 3:50:51 PM