Page 1 | © 2014 by Janice L. Epstein 5.4 Optimization 5.4: Optimization
Example: A company wishes to design a rectangular box with a square
base and no top. The volume of the box is 27 cubic inches. The cost per
square inch of the bottom is twice the cost of the sides. What are the
dimensions of the box that has the minimum cost?
Example: A poster is to have an area of 180 in2 with 1 in margins on the
sides and bottom and 2 in margin on the top. What dimensions will give
the largest printed area?
Page 2 | © 2014 by Janice L. Epstein 5.4 Optimization Example: Find the point on the line y = 2 x - 3 that is the closest to (-1,3)
3
2
1
-1
1
2
3
-1
-2
-3
Example: Find the dimensions of the rectangle of the largest area that has
its base on the x-axis and the other two vertices above the x-axis lying on
the parabola y = 8 - x 2
9
8
7
6
5
4
3
2
1
-3
-2
-1
1
2
3
Page 3 | © 2014 by Janice L. Epstein 5.4 Optimization Example: Pipe needs to be laid connecting A with B. The cost along the
level stretch A to C is $10 per foot. The cost along the diagonal stretch
from C to B is $30 per foot. At what point C will the cost of laying the
pipe be minimized?
B
30’
D
C
40’
A
Page 4 | © 2014 by Janice L. Epstein 5.4 Optimization Example: A piece of wire 10 m long is cut into two pieces. One piece is
bent into a circle and the other piece is bent into a square. How should the
wire be cut so that the total area enclosed is minimized? How should the
wire be cut to maximize the total area enclosed?