1. 4.4 Optimization Buffalo Bill’s Ranch, North Platte, Nebraska
Created by Greg Kelly, Hanford High School, Richland, Washington
Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Photo by Vickie Kelly, 1999
Buffalo Bill’s Ranch, North Platte, Nebraska

2. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
domain: x > 0 and x > 0
x < 20
There must be a maximum at this critical value (the area at the endpoints = 0).

3. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

4. To find the maximum (or minimum) value of a function:
Write the OPTIMIZED function, restate it in terms of
one variable, determine a sensible domain.
Find the first derivative, set the derivative to zero/
undefined, find function value at those critical points.
3 Find the function value at each end point.

5. We can minimize the “material” by minimizing the can’s surface area.
Example 5:
What dimensions for a one liter cylindrical can will use the least amount of material?
Motor
Oil
We can minimize the “material” by minimizing the
can’s surface area.
To rewrite using one variable, we need another equation that relates r and h:
area of
“lids”
lateral
area

6. Example 5:
What dimensions for a one liter cylindrical can will use the least amount of material?
area of
ends
lateral
area

7. p Reminders: If the function that you want to optimize has more than
one variable, find a second connecting equation and substitute to rewrite the function in terms of one variable.
Evaluate the function for that critical input value.
If a domain endpoint could be the maximum or minimum, remember to evaluate the function at each endpoint, too. p