You want to construct a can that holds 150 cubic inches of
juice as cheaply as possible. The top and bottom costs .1 cents per square
inch and the side costs .09 cents per square inch. What should the dimensions
of the can be?
Solution
We use the volume formula to get
150 = pr^{2}h
and next calculate the cost
Cost = 2pr^{2}(0.1)
+ 2prh(0.09)
Cost of top and bottom + Cost of sides
The volume equation gives us:
150
h =
p r^{2}
so that
150
C = 0.2pr^{2} +
0.18pr
p r^{2}
27
= 0.2pr^{2 }+
r
To find the minimum cost we take the derivative and set it equal to 0:
27
C' = 4pr - ^{
}= 0
r^{2}
So that
4pr^{3}
= 27
or
27
r^{3} =
4p
r = 2.14 in
so that
150
h =
= 10.4 in
p(2.14^{2})
The can should be constructed so that its radius is 2.14
inches and its height is 10.4 inches.