- 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 in3/min
Want: dr/dt when r = 2 in.
Note: decreasing quantity means a negative derivative.
- 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
- 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 r3, so
(4/3) pi (3 r2 dr/dt)
= 4pi r2 dr/dt
Warning: don't forget the chain rule!
- Solve for the rate you want. For example:
dr/dt = (dV/dt) / (4pi r2)
- 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.