3.7 Optimization
Problems
Solve real-life optimization problems
Standard 4.5a
A family plans to fence in a rectangular patio area
behind their house. They have 120 feet of fence to
use. What would be the dimensions of the
rectangular region if they wanted to make the patio
area enclosed as large as possible?
y
x
Dimensions 30 ft. x 30 ft.
A strip of metal 20 inches wide is going to be bent
to form an open gutter of rectangular cross section.
What dimensions of the cross section will make the
carrying capacity maximum?
x
y
when x = 5,
y = 20 – 2(5) = 10
dimensions 5 in. x 10 in.
A one foot square of metal is to be made into a box
without a top by cutting equal squares out of the
corners and turning up the edges. What dimensions
will make the volume maximum? What is the
maximum volume?
x
x
1 – 2x
x
x
1 – 2x
1 – 2x
x
1 – 2x
x
x
1 – 2x
1 – 2x
x
x
maximum
dimensions: 1/6 x 2/3 x 2/3
minimum
volume: 2/27
Express 10 as the sum of two numbers whose product
is maximum.
x = 5, y = 5
Express 9 as the sum of two positive numbers such that
the product of one by the square of the other is a
maximum.
Minimum
Maximum
x = 6, y = 3
A page is to contain 30 square inches of print. The
margins at the top and bottom of the page are each 2
inches wide. The margins on each side are 1 inch wide.
What dimensions will minimize the amount of paper
used?
2
1
y
x
2
1
Minimum
Determine the dimensions of a rectangular solid
(with a square base) with maximum volume if its
surface area is 337.5 square centimeters.
S  2 x  4 xy  337 .5
2
y
y


x
x

337 .5  2 x
2
4x
2


2 337 .5  2 x
2
V  x y  x 

4
x


V  84 .375 x 

1
2
x
3
V  84 .375 x 
dV
 84 .375 
dx
84 .375 
3
2
1
2
3
2
x
3
x
2
x  0  x  56 .25
2
2
 x  7.5
2
d V
dx
2
 3 x  0 for

x  7.5

The maximum value occurs when x = 7.5 cm and
y = 7.5 cm.