CHAPTER 3SECTION 3.7

OPTIMIZATION PROBLEMS

Applying Our Concepts

• We know aboutmax and min …

• Now how can we use thoseprinciples?

Use the Strategy• What is the quantity to be optimized?

– The volume

• What are the measurements (in terms of x)?

• What is the variable which will manipulated to determine the optimum volume?

• Now use calculus principles

x

30”

60”

Guidelines for Solving Applied Minimum and Maximum Problems

Optimization

Optimization

Maximizing or minimizing a quantity based on a given situation

Requires two equations:

Primary Equation

what is being maximized or minimized

Secondary Equation

gives a relationship between variables

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.

1. An open box having a square base and a surface area of 108 square inches is to have a maximum volume. Find its dimensions.

1. An open box having a square base and a surface area of 108 square inches is to have a maximum volume. Find its dimensions.

Primary Secondary2V x y

Intervals: 0,6 6, 108Test values: 1 10

V ’(test pt) V(x) inc dec

rel max

6x

xx

y2108 4x xy

2108

4

xy

x

22 108

( )4

xV x x

x

3108

4 4

x x

31427x x

234'( ) 27V x x

23427 0x

2 4327x

6x

Domain of x will range from x being as small as possible to x as large as possible.Smallest

(x is near zero)

Largest

(y is near zero)2 108x 0x

2 4SA x xy

21 8 6

6

0

4y

3 y

Dimensions: 6 in x 6 in x 3 in

2. Find the point on that is closest to (0,3). 2f x x

2 20 3d x y

2. Find the point on that is closest to (0,3).Primary Secondary

Intervals: 52, 5

2 0,Test values: 3 1d ’(test pt)

d(x) dec increl min

52x

2f x x

2y x

2 23d x y

***The value of the root will be smallest when what is inside the root is smallest.

2 23d x y

22 2 3d x x x 2 4 26 9d x x x x

4 25 9d x x x

3' 4 10d x x x 34 10 0x x

22 2 5 0x x 20 2 5 0x x

520x x

520,

1

decrel max

52 ,3inc

rel min

52x

5 52 2, 5 5

2 2,

Minimize distance

2. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the print be to use the least paper?

2. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the print be to use the least paper?

Primary Secondary

2 3A x y 24xy

11

1.5

1.5

224 in

x

2x

y3y 24

xy 24( ) 2 3xA x x 4824 3 6xx 13 48 30x x

248'( ) 3x

A x 2

2

3 48x

x

crit #'s: 0, 4x Intervals: 0,4 4,24Test values: 1 10

A ’(test pt) A(x) incdec

rel min

4x

Smallest

(x is near zero)

Largest

(y is near zero)

24x 0x

44

2y

6y

Print dimensions: 6 in x 4 in

Page dimensions: 9 in x 6 in

1. Find two positive numbers whose sum is 36 and whose product is a maximum.

P xy

1. Find two positive numbers whose sum is 36 and whose product is a maximum.

Primary Secondary

36x y 36y x 36P x x x

'( ) 36 2P x x 36 2 0x

18x

36 18 18y

Intervals: 0,18 18,36Test values: 1 20P ’(test pt)

P(x) inc decrel max

18x

18,18

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

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.

Example #1

• A company needs to construct a cylindrical container that will hold 100cm3. The cost for the top and bottom of the can is 3 times the cost for the sides. What dimensions are necessary to minimize the cost.

r

h 2

2

22 rrhSA

hrV

Minimizing Cost

hr

hr

2

2

100

100

2

22

2200

2100

2

rr

SA

rr

rSA

26200

)( rr

rC

rr

rC

12200

)(2

2

2

22 rrhSA

hrV

Domain: r>0

Minimizing Cost

rr

12200

02

rr

rC

12200

)(2

312200

3

2

12200

12200

r

r

rr

744.13350 r

0)744.1(

12100

)(3

Cr

rC

Concave up – Relative min

464.10

1002

h

hr

The container will have a radius of 1.744 cm and a height of 10.464 cm

1.7440

- - - - - - + + + + +

C' changes from neg. to pos. Rel. min