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 40x 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' 12 x10 1

2 x1x2

1x4

3 / 21y" x4

1y" 04

therefore a max

12

1 and 192

2. 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 800xA' 4x 8000 4x 800x 200

A" 4 Therefore a max

A 200 2 200 800 80000

3. 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' 22 5 x

x0 25 x

x 2 5 x

x 4 5 x

x 4x 2, 2

2

+ _

Therefore x = 2 is a max

2 22 y 5y 1

2x y 2 2 1 5

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

x

y h

2 2 2

2

x y h1V x y3

2 21V h y y

3

2 3V y1 1h3

y3

2

2

2

2

V ' y

0

1 h3

1 h3

y

y h3

221 h3

2

h

V ' y

V " y

h2 0"3 3

V

22 2 2 2 2 h 2 2 hx h y x h x h h

3 33 3

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

r

0.5h R

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

22 2 hV r h V R h

4

32 hV R h

4

22

22

3 hV ' R4

3 h R4

2Rh3

3 hV "2

2R32R 3V " 0

23

Therefore a max

22 21r h R

2

34 RV3 3

6. 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 xy 4y 8x 32

100 100x 4 8x 32x x

400132 8xx

2

400A' 8x

2

4000 8x

x 50

y 2 50

3

200A" 0x

therefore a minimum

4 50 8 2 50

7. 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 4xx

2

2

2

1024S xx

1024S' 2xx

10240 2xx

x 8

y 4

3

2048S" 2 0x

therefore a min

8 x 8 x 4

8. Find the point on the curve that is a minimum distance from the point (4, 0).

y x

2 22D x 4 y 0

y x

2 2 2D x 8x 16 y 2 2D x 7x 16

Now let L = D^2. We can do this since the minimum of D^2would be the same as the minimum of L.

2L x 7x 16L' 2x 70 2x 7

7x2

7y2

L" 2 0

Therefore a minimum

7 7,2 2

9. Suppose that the revenue of a company can be represented with the function r(x) = 48x, and the company’s cost function is , where x represents thousands of units and revenue and cost are represented in thousands of dollars. What production level maximizes profit, and what is the maximum profit to the nearest thousand dollars?

3 2c x x 12x 60x

dP dr dcP r cdx rx dx

r 48xr ' 48

3 2

2

c x 12x 60x

c ' 3x 24x 60

2dP 3x 24x 12

dx

20 3x 24x 12 x 0.54, 7.46

At 7.46, r = 358.08, c = 194.34, or P = 163.14At 0.54, r = 25.92, c = 29.06 or P = -3.14

Since 163.14 > -3.14, max profit is $163,000 which occurs when7460 units are made