A storage vessel is constructed by attaching a hemispherical cap to the top of a right circular cylindrical body that shares the same radius. The vessel must contain exactly 18π ft3 of liquid when completely filled.
The manufacturer uses two grades of sheet metal. The curved lateral wall of the cylindrical portion is made from standard sheet metal costing \2.00persquarefoot,whilethehemisphericalcapismadefromcorrosion−resistantsheetmetalcosting3.00 per square foot. (The circular base of the cylinder is not present; the vessel is intended to sit directly on the ground.)
Assume all seams have negligible width and waste material can be ignored.
(a) Let r be the common radius (in feet) of both the cylinder and the hemisphere, and let h be the height (in feet) of the cylindrical portion. Write an expression for the total manufacturing cost C(r,h) in terms of r and h. Your expression should incorporate both the required volume condition and the different material prices.
(b) Express h solely as a function of r by using the fixed-volume requirement, and rewrite C as a function C(r) of a single variable.
(c) Determine the radius r that minimizes the total manufacturing cost, and find the corresponding height h. Provide exact values (in terms of rational multiples of π where appropriate).
(d) Verify that the critical point you found in part (c) yields a minimum total cost rather than a maximum or point of inflection.
(e) Interpret your result in practical terms: state the minimum possible manufacturing cost (in dollars, to the nearest cent) and describe how the relative prices of the two metals influence the optimal dimensions.