Warmup- no calculator1)

2)

4.4: Modeling and Optimization

ex. Find two numbers who’s sum is 20 and product is as largeas possible

Steps:1) find a primary (what your optimizing)

and secondary equation (concrete info in problem)2) solve the secondary for one variable3) substitute it into the primary4) find extrema of the functioncheck endpoints and critical #’s

Keys to Optimization

primary: f(x,y) = xy

secondary: x + y = 20

y = -x+20

f(x)= x(-x+20) f(x) = -x2+20x

202)( xxf =0x = 10, so y = 10

Find your: primary equation (idea your optimizing) secondary equation (additional info in problem)

.

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 the endpoints are minimums.

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 it in terms of one variable.

2 Find the first derivative and set it equal to zero.

3 Check the end points if necessary.

Ex.

A farmer plans to fence a rectangular pasture adjacent to a river.The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions wouldrequire the least amount of fencing if none is needed along the river?

Four feet of wire is to be used to form a square and a circle.How much wire should be used for the square and howmuch should be used for the circle to enclose a maximum area?

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 area.

22 2A r rh area ofends

lateralarea

We need another equation that relates r and h:

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

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 the end points could be the maximum or minimum, you have to check.

Notes:

If the function that you want to optimize has more than one variable, use substitution to rewrite the function.

If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check.

the end