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. 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:

• 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) 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