4.4 Optimization Buffalo Bills Ranch, North Platte, Nebraska Created by Greg Kelly, Hanford High...

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.