Ch 4.4 Modeling and OptimizationGraphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy

Example

An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20 by 25 inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the maximum volume?

xx

25

20

Example

An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20 by 25 inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the maximum volume?

xx

25

20

2

3 2

2

V = 25 2x 20 2x x, 0 < x < 10

= 500 - 90x + 4x x

= 4x - 90x + 500x

V x = 12x - 180x + 500 = 0 at x 3.68, 11.32

Test endpoints and critical points:

V 0 = 0

V 3.68 = 820.53 Max volume

2

at x = 3.68

V 10 = 0 Volume = 820.53 in

+ -

3.68

V '

Example

Find two numbers whose sum is 20 and whose product is as large as possible.

Example

Find two numbers whose sum is 20 and whose product is as large as possible.

2

x + y = 20, y = 20 - x, 0 < x 20

P x = x y

P x = x 20 x

= 20x - x

P x = 20 - 2x = 0 at x = 10

Test endpoints and critical point:

P 0 = 0

P 10 = 100 The 2 numbers are 10,10

P 20

= 0

+ -

10

P '

Example

What dimensions should a right cylinder containing a one liter of oil be that would use the least amount of material? (1 liter = 1000 cm3.

r

h

Example

What dimensions should a right cylinder containing a one liter of oil be that would use the least amount of material? (1 liter = 1000 cm3.

r

h

22

2A

22

2A

A 2

A A3

1000V = r h = 1000 h =

r

S = 2 r + 2 r h

1000 = 2 r + 2 r

r

2000S = 2 r +

r

2000S = 4 r - = 0 at r 5.42

r4000

S = 4 + S 5.42 > 0 minr

Least

materials at r 5.42 cm and h 10.84 cm