A food company is producing an open‐top cylindrical can that must hold 1.0 liter of…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A food company is producing an open‐top cylindrical can that must hold 1.0 liter of soup (1,000 cm

^{3}

). The side of the can is manufactured from sheet metal costing

0.02 per cm

^{2}

, while the circular base is stamped from a thicker metal costing

0.05 per cm

^{2}

. The company wants to minimize the total manufacturing cost per can. (a) Let

r

be the radius (in cm) and

h

be the height (in cm) of the can. Write an expression for the total cost

C

(in dollars) of one can in terms of

r

and

h

. (b) Use the fixed‐volume requirement to express

h

as a function of

r

. (c) Rewrite the cost function

C

solely in terms of

r

, and state an appropriate domain for

r

. (d) Determine the radius and height that minimize the manufacturing cost. Then calculate the minimum cost per can, rounded to the nearest cent. (e) Discuss how sensitive the total cost is to small changes in the optimal radius and what this implies for production tolerances.

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