- 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:
Given:

*dV*/*dt*= 8 in^{3}/min

Want:*dr*/*dt*when .*r*= 2 inNote: decreasing quantity means a negative derivative.

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

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

- Solve for the rate you want. For example:
*dr*/*dt*= (*dV*/*dt*) / (4*pi**r*^{2})

- Evaluate using the given rates and other values. For example:
*dr*/*dt*= 8 / (4 *pi*2^{2})= (8/16 *pi*)= 1/(2 *pi*)

So*dr*/*dt*is approximately 0.16 in/min.

Comments to: dpvc@union.edu

Created: Oct 14 1997 --- Last modified: Oct 27, 1998 3:50:51 PM