[Note for Mac users] |

Question: Suppose air is blown into a spherical balloon at a rate of ^{3}/sec |

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

- Draw a picture of the situation.

- 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!

- Write down explicitly what rates are involved. For example:
We know that

*dV*/*dt*= 36 in^{3}/sec.

We want*dr*/*dt*when .*r*= 3 inNote:

*decreasing*quantity means a*negative*derivative.

- Find an equation relating the quantities involved. For example:
.*V*= (4/3) p*r*^{3}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

- Differentiate both sides with respect to
*t*(like implicit differentiation), considering the changing quantities as functions of*t*. For example:*V*= (4/3) p*r*^{3}, so

*dV*/*dt*=(4/3) p (3 *r*^{2}*dr*/*dt*)= 4 p r ^{2}*dr*/*dt***Warning:**don't forget the chain rule!

- Substitute the given rates and other constants. For example:
36 = 4 p 3

^{2}*dr*/*dt***Warning:**this comes*after*differentiating!

- Solve for the desired rate. For example:
*dr*/*dt*= 36 / (4 p 3 ^{2})= (36/36p) = 1/p .

So*dr*/*dt*is approximately 0.32 in/sec.

Comments to: dpvc@union.edu

Created: Oct 14 1997 --- Last modified: Mar 1, 1999 12:07:48 PM