Notes on Related Rates:

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:

   Given: dV/dt = 8 in³/min
   Want: dr/dt when r = 2 in.

   Note: decreasing quantity means a negative derivative.

   
   

4. 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 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) pi r³, so
   dV/dt = (4/3) pi (3 r² dr/dt) = 4pi r² dr/dt

   Warning: don't forget the chain rule!

   
   

6. Solve for the rate you want. For example:

   dr/dt = (dV/dt) / (4pi r²)

   
   

7. 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.