4.4 Optimization

Buffalo Bill’s Ranch, North Platte, NebraskaCreated by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Photo by Vickie Kelly, 1999

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?

x x

40 2x

40 2A x x

240 2A x x

40 4A x

0 40 4x

4 40x

10x 40 2l x

w x 10 ftw

20 ftl

There must be a local maximum here, since thearea at the endpoints = 0.

domain: x > 0 and 40 - 2x > 0 x < 20

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?

x x

40 2x

40 2A x x

240 2A x x

40 4A x

0 40 4x

4 40x

10x

10 40 2 10A

10 20A

2200 ftA40 2l x

w x 10 ftw

20 ftl

To find the maximum (or minimum) value of a function:

1 Write the OPTIMIZED function, restate it in terms of one variable, determine a sensible domain.

2 Find the first derivative, set it equal to zero/ undefined, find function value at those critical points.

3 Find the function value at each end point.

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material?

We can minimize the material by minimizing the can’s surface area.

22 2A r rh area of“lids”

lateralarea

To rewrite using one variable, we need another equation that relates r and h:

2V r h

31 L 1000 cm3 2

cm1000 r h

2

1000h

r

22

10 02

02A r r

r

2 20002A r

r

2

20004A r

r

2liter1 r h

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material?

22 2A r rh area ofends

lateralarea

2V r h

31 L 1000 cm21000 r h

2

1000h

r

22

10 02

02A r r

r

2 20002A r

r

2

20004A r

r

2

20000 4 r

r

2

20004 r

r

32000 4 r

3500r

3500

r

5.42 cmr

2

1000

5.42h

10.83 cmh

If a domain endpoint could be the maximum or minimum, you have to evaluate the function at each endpoint, too.

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.

To confirm that the critical value you’ve found is a maximum or minimum, you should evaluate the function for that input value.