Mathematics

Problem

A beverage company plans to manufacture an open-top cylindrical can that must hold exactly

1000\text{ cm}^3

of liquid. The lateral surface is made from aluminum that costs \

0.03 per square centimeter, while the circular base is stamped from a reinforced alloy that costs

0.07 per square centimeter. Let

r

denote the interior radius (in centimeters) of the can and

h

its interior height. Answer the following, showing all reasoning. (a) Express the total material cost

C

as a function of

r

only by eliminating

h

. (b) Compute

\dfrac{dC}{dr}

. (c) Find all critical numbers of

C

for

r>0

and give their numerical values (to three decimal places if necessary). (d) Verify that the critical number you found yields the minimum possible material cost, and state the corresponding cost. (e) Determine the exact optimal dimensions

r

and

h

(include appropriate units) and interpret what these values mean for the manufacturer.

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