Section 5.4Using Calculus to Solve Optimization Problems

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5.3

1. The sum of two nonnegative numbers is 20. Find the numbers (a) if the sum of their squares is to be as large as possible.

Let the two numbers be represented by x and 20 – x.

22y x 20 x

y ' 2x 2 20 x 1

0 4x 40

x 10

f " x 4 makes x = 10 a minimum.

Maximum must occur at an endpoint.

0 and 20

1. The sum of two nonnegative numbers is 20. Find the numbers (b) If the product of the square of one number and the

cube of the other is to be as large as possible

23y x 20 x

Let the two numbers be represented by x and 20 – x.

22 3

2

2

y ' 3x 20 x 2x 20 x 1

y ' x 20 x 3 20 x 2x

0 x 20 x 60 5x

x 0, 20,12

12 20

+ +_

Max at 12, Min at 20

12 and 8

1. The sum of two nonnegative numbers is 20. Find the numbers (c) if one number plus the square root of the other is as large as possible.

Let the two numbers be represented by x and 20 – x.

y 20 x x

1y' 1

2 x1

0 12 x

1x

21

x4

3 / 21y" x

41

y" 04

therefore a max

34

1and19

4

2. A rectangular pen is to be fenced in using two types of fencing. Two opposite sides will use heavy duty fencing at $3/ftwhile the remaining two sides will use standard fencing at $1/ft.What are the dimensions of the rectangular plot of greatest areathat can be fenced in at a total cost of $3600?

3x

1y

A xy 2 3x 2 1y 3600 3x y 1800

A x 1800 3x 2A 1800x 3x

A ' 1800 6x 0 1800 6x

300 x

A " 6Therefore max

900 y

The dimensions of a rectangular plot of greatest area are 300 x 900

3. A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m of fence at your disposal, what is the largest area you can enclose?

2

2x y 800 y 2x 800

A xy A x 2x 800 A 2x 800x

2A 2x 800x

A' 4x 800

0 4x 800

x 200

A " 4 Therefore a max

A 200 2 200 800 80000

The largest area you can enclose is 80000

x xy

4. An open-top box with a square bottom and rectangular sides is to have a volume of 256 cubic inches. Find the dimensions that require the minimum amount of material.

2

2

S x 4xy

V x y 256

2

2

256S x 4x

x

2

2

2

1024S x

x1024

S' 2xx

10240 2x

xx 8

y 4

3

2048S" 2 0

x

therefore a min

8 x 8 x 4

yx

x

5. Find the largest possible value of 2x + y if x and y are the lengths of the sides of a right triangle whose hypotenuse is

units long.5

2 2

B 2x y

x y 5

2B 2x 5 x

2

2

2

2 2

2

2xB' 2

2 5 xx

0 25 x

x 2 5 x

x 4 5 x

x 4

x 2, 2

2+ _

Therefore x = 2 is a max

2 22 y 5

y 1

2x y 2 2 1 5

x

y 5

6. A right triangle of hypotenuse 5 is rotated about one of its legs to generate a right circular cone. Find the cone of greatest volume.

x

y 5

2 2 2

2

x y 5

1V x y

3

21

V 25 y y3

3V y25 1

3y

3

2

2

V ' y

0

25

325

3

y5

3

y

225V ' y

3V " 2 y

5 5V " 2 0

3 3

2

2 2 2 5 2 2 5x 25 y x 25 x 5 5

3 33 3

Therefore max

7. Determine the area of the largest rectangle that may be Inscribed under the curve xy e on x, x

A 2xyxA 2xe

x xA ' 2e 2xe

x

2A ' (1 x)

e

x

20 (1 x)

e

x 1

1

_+

Therefore max

1 2A 2 1 e

e

8. (calculator required) A poster is to contain 100 square inches of picture surrounded by a 4 inch margin at the top and bottom and a 2 inch margin on each side. Find the overall dimensions that will minimize the total area of the poster.

A PIC xy 100

A POS x 4 y 8

A POS y 8 x 4

1008 x 4

x

400132 8x

x

y 14.142

Since f’ changes from neg to pos,

we have a minimum

11.1071 22.142

4

4

2 2x

y

9. Determine the point on the graph of y

nearest

x 2 that is

to the point (5, 0).

2 2D x 5 y 0

1/ 22D x 10x 25 x 2

1/ 22D x 9x 23

1/ 221D' x 9x 23 2x 9

2

2

2x 9D'

2 x 9x 23

9x

2 9/2

+_

Therefore min9 5

,2 2

210. The graphs of y 25 x , x 0 and y 0 bound a region

in the first quadrant. Find the di

maximum peri

mensions of the rectangle of

that can be inscribed in thismete region.r

P 2x 2y 2P 2x 2 25 x

1/221P' 2 2 25 x 2x

2

2

2xP' 2

25 x

22x 2 25 x

2 2x 25 x 2 25

x2

+_

Therefore max

5

2

5 5

2 25

x2

11.(calculator required) Find the dimensions of the rectanglewith maximum area that can be inscribed in a circle of radius 10.

A 4xy2 2x y 100

x 7.071 y 7.071

14.142 14.142

Since f’ changes from pos to neg,

we have a maximum

2A 4x 100 x

3 2

12. Suppose that the revenue of a company can be represented with the function

r x 48x, and the company's cost function is c x x 12x 60x, where x

represents thousands of units and revenue and cost ar

e represented in thousands

of dollars. What maximizes profit and what is the maximum profit

to the nearest th

prod

ousa

uction lev

nd do

el

llars?

P r x c x

3 2P 48x x 12x 60x 3 2P 12x x 12x

2dP3x 24x 1

dx2

20 3x 24x 12

x 0.54, 7.46

At 7.46, r = 358.08, c = 194.34, or P = 163.14

Max profit is $163,000 which occurs when 7460 units are made

Max at x = 7.46 since f ‘

changes from pos to neg

13. A tank with a rectangular sides is to be open at the top. It is to be constructed so that its width is 4 m and its volume is 36 cubic meters. If building the tank costs $10 per square meter for the base and $5 per squaremeter for the sides, what is the cost of the least expensive tank?

yx

44xy 36 xy 9

C $10 4y $5 2 4 x $5 2 x y

9C 40 40x 10 9

x

2C' 360x 40

2

3604 x0

x3 y 3

3

720C" 0

xTherefore min

39

C $3340 403

10 9 0

x14. Find the minimum distance from the origin to the curve y e

2 2D x 0 y 0

1/ 22 2xD x e

CALCULATOR REQUIRED

x 0.426

Minimum since f ‘ (x) changes fromneg to pos at –0.426

1/ 22 2 0.426D 0.426 e 0.780

2

2

15. (calculator required) Consider f x 12 x for 0 x 2 3. Let

A(t) be the area of the triangle formed by the coordinate axes and the

tangent to the graph of f at the point t,12 t . For what value of t is

A(t) a minimum?

A t1

xy2

f ' x 2x f ' t 2t

2y 12 t 2t x t

2 2, y 12 tI 2f x 0 0 y t 2t t 1

22

, 12 t 2tt 1

xIf y 0 02

xt

t2

2222

t 121 t 12t 12

2 2t 4t

Since A ‘ changes from neg to pos, min area at t = 2

2

16. Find the maximum distance measured horizontally between

the graphs of f x x and g x x for 0 x 1.

D y y

1D' 1

2 y

1 11 2 y 1 4y 1 y

42 y

+_

Therefore max

1

41 1 1

D4 4 4

17. (calculator required) What is the area of the largest rectangle

that can be inscribed under the graph of y 2cos x for x ?2 2

A 2xy A 4xcos x

Since A ‘ changes frompos to neg at x = 0.860,

max of A occurs at x = 0.860

A 4 0.86033358 cos 0.86033358 2.244

18.Consider the set of all right circular cylinders for which thesum of the height and diameter is 18 inches. What is the radiusof the cylinder with the maximum volume?

2

h 2r 18

V r h

h 18 2r 2V r 18 2r

2 3V 18 r 2 r 2V ' 36 r 6 r

V ' 6 r 6 r r 0, 6

V " 36 12 r

V " 6 0 max

X

2

19. An isosceles triangle has one vertex at the origin and the other

two at the points where a line parallel to and above the x-axis

intersects the curve f x 12 x . Find the maximum area of the triangle .

A xy 2 3A x 12 x A 12x x

2A ' 12 3x 212 3x

x 2

A " 6x

A " 2 0

max

2f 2 12 2 8

A xy 2 8 16

2

20. Find the height of the rectangle with largest area that can be

1inscribed under the graph of y

1 2x

2

2xA 2xy A

1 2x

2

22

2 1 2x 4x 2xA '

1 2x

22

2 22 2

2 1 2x2 4xA '

1 2x 1 2x

__

1

2

1

2

+

1maxat

2

2

1 1y

211 2

2

3 2

21. Find the minimum distance from the origin to the curve

x 3x 4x 12y

6

23 22 x 3x 4x 12

D x 0 06

CALCULATOR REQUIRED

Two possibilities whereD ‘ changes from negto pos, denoting a min

D 1.0696716 1.410

D 1.781107 1.899

22. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 5.

r

0.5h R

h – height of cylinderr – radius of cylinderR – Given radius of sphere

2V r h

3hV 25 h

4

23 hV ' 25

4

3 hV "

2

103

10 3V " 023

22 21

r h 52

Therefore a max500

V3 3

22 h

V 5 h4

23 h25

4

10h

3