Calculus 127 Quiz 6 Solutions

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 tex2html_wrap_inline56 cm then the cost function C(x,h) is:

(a) tex2html_wrap_inline58
(b) tex2html_wrap_inline60
(c) tex2html_wrap_inline62
(d) tex2html_wrap_inline64
(e) tex2html_wrap_inline66
(f) none of the above

Answer: (c) tex2html_wrap_inline68 .

2) (6 points) Find the dimensions which minimize the cost.

(a) 2.5 cm tex2html_wrap_inline70 2.5 cm tex2html_wrap_inline70 108 cm
(b) 4.5 cm tex2html_wrap_inline70 4.5 cm tex2html_wrap_inline70 33.3 cm
(c) 6.5 cm tex2html_wrap_inline70 6.5 cm tex2html_wrap_inline70 16 cm
(d) 7.5 cm tex2html_wrap_inline70 7.5 cm tex2html_wrap_inline70 12 cm
(e) 8 cm tex2html_wrap_inline70 8 cm tex2html_wrap_inline70 10.25 cm
(f) 10 cm tex2html_wrap_inline70 10 cm tex2html_wrap_inline70 6.75 cm
(g) none of the above

Answer: (d)

Step 1: The constraint equation is

displaymath52

Step 2: Use the constraint equation in order to express the cost function as a function of one veriable (say x). Since tex2html_wrap_inline94 , we get that the cost, as a function of x, is

displaymath53

Step 3: Minimize the cost by solving

eqnarray40

We can verify that x=7.5 is the absolute minimum by using the first derivative test: C'(x) is negative in the interval tex2html_wrap_inline98 and positive in the interval tex2html_wrap_inline100 .