A manufacturer designs right circular cans that must hold exactly of liquid
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A manufacturer designs right circular cans that must hold exactly 1000π cm3 of liquid. Let r (in cm) denote the radius of the base and h (in cm) denote the height. The exterior surface area of a can with a lid is S(r,h)=2πr2+2πrh. The company wishes to minimize the ratio 1000πS(r,h) so that the amount of material used per unit volume is as small as possible.
(a) Use the fixed–volume requirement to write h explicitly as a function of r. Then express the ratio 1000πS(r,h) purely in terms of r. Simplify your result.
(b) Determine all positive critical values of r for the function obtained in part (a). Show all differentiation steps and clearly state any domain restrictions.
(c) Use either a first–derivative sign chart or a second–derivative computation to justify which critical value yields the minimum ratio. State the minimizing radius (in cm) to three significant figures.
(d) For the optimal can found in part (c), compute the corresponding height and the minimized material–to–volume ratio. Include appropriate units in your conclusion.