Optimization
5.4
A box with a square base and open top must have a volume of
32,000 cubic cm. Find the dimensions of the box that minimize
the amount of material used.
40 by 40 by 20
Find two numbers whose sum is 40 and whose product is as
large as possible.
20 and 20
What is the smallest perimeter possible for a rectangle whose
area is 25 square inches?
P=20
The top and bottom margins of a poster are each 2 cm and
the side margins are each 1 cm. If the area of printed material
on the poster is fixed at 92 sq. cm, find the dimensions of the
poster with the smallest area.
x  46
8.782 by 17.565
An advertisement consists of a rectangular printed region plus
1-in margins on the sides and 1.5-in margins at the top and
bottom. If the total area of the advertisement is to be 120 sq
in, what dimensions should the advertisement be to maximize
the are of the printed region?
A two-pen corral is to be built. The outline of the corral forms
two identical adjoining rectangles. If there is 120 ft of fencing
available, what dimensions of the corral will maximize the
enclosed area?
What dimensions for a one liter cylindrical can will
use the least amount of material?
We can minimize the material by minimizing the area.
We need another
equation that
relates r and h:
V   r 2h
3
1
L

1000
cm


1000   r 2 h
1000
h
2
r
A  2 r 2  2 rh
area of
ends
lateral
area
1000
A  2 r  2 r 
 r2
2
2000
A  2 r 
r
2
2000
A  4 r  2
r
What dimensions for a one liter cylindrical can will
use the least amount of material?
A  2 r 2  2 rh
V   r 2h
3
1
L

1000
cm


1000   r 2 h
1000
h
2
r
1000
  5.42 
2
area of
ends
lateral
area
1000
2
A  2 r  2 r 
 r2
2000
 4 r
2
r
2000  4 r 3
500

2000
A  2 r 
r
2
h
h  10.83 cm
 r3
500
2000
A  4 r  2
r
r
2000
0  4 r  2
r
r  5.42 cm
3


What are the dimensions of the largest rectangle that
can be inscribed in the ellipse x2  4 y 2  16 ?
x2  4 y 2  16
16  x 2
y
2
pg 214 #2,5,6,21(calc)