Notes on Related Rates:

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.