OPTIMIZATION PROBLEMS

Procedures for Solving Optimization Problems

1. Assign symbols to all given quantities and quantities to be determined.  When feasible, make a sketch.
2. Decide what is being maximized or minimized and write an equation representing that quantity.
3. Try to reduce the equation to one having only one variable. Make equations of other relationships to help do this.
4. Write down the domain of the equation to be used. There may be endpoints to consider.
5. Determine the maximum or minimum value by using the derivative of the stated equation.

Example
Determine the dimensions of a box of maximum volume that can be made from a piece of material 8" by 10". The box is to be made by cutting square pieces from the corners. The box will NOT have a top.

1. Draw a picture of a piece of material 8" by 10".  Show the squares in each corner that are to be cut out and label the edge of each square with x.  Now, label the length of the box
   10 - 2x, the width of the box 8 - 2x, and the height of the box x.
2. The volume is to be maximized, so
              V = lwh
3.            V = (10 - 2x)(8 - 2x)x
4. 0 < x < 4 because the square being cut out can be no longer than half the width
5. V = 80x - 36x² + 4x³
   dV/dx = 80 -72x + 12x²
           0 = 3x² - 18x + 20
   Since this cannot be factored, use the quadratic formula or a calculator to find the roots.
             x = 1.472, 4.528
   Since 0 < x < 4, we can throw out 4.528.
   
                           +                        -                        
       dV/dx   _____________|___________
                           /             1.472   \
   
   The line test shows there will be a maximum at x = 1.472 because dV/dx changed from positive to negative there.
6. The dimensions of the box will be
             length = 10 - 2(1.472) = 7.055"
             width = 8 - 2(1.472) = 5.055"   
             height = 1.472"

Problems