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:

• Geometric formulas (like the one above)
• Trigonometric formulas (you have an angle and a triangle's side in your problem)
• The Pythagorean Theorem
• Similar triangles

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