1) (4 points) A rectangular box with a square base and a volume of
675 cubic centimeters is to be constructed.
The sides cost $1.25 per square centimeter and
the top and bottom cost $2.00 per square centimeter.
If the dimensions of the box are
cm
then the cost function C(x,h) is:
(a)
(b)
(c)
(d)
(e)
(f) none of the above
Answer: (c)
.
2) (6 points) Find the dimensions which minimize the cost.
(a) 2.5 cm 2.5 cm
108 cm
(b) 4.5 cm 4.5 cm
33.3 cm
(c) 6.5 cm 6.5 cm
16 cm
(d) 7.5 cm 7.5 cm
12 cm
(e) 8 cm 8 cm
10.25 cm
(f) 10 cm 10 cm
6.75 cm
(g) none of the above
Answer: (d)
Step 1: The constraint equation is
Step 2:
Use the constraint equation in order to
express the cost function as a function of one
veriable (say x). Since , we get that the cost,
as a function of x, is
Step 3: Minimize the cost by solving
We can verify that x=7.5 is the absolute minimum by using the first
derivative test: C'(x) is negative in the interval
and positive in the interval
.