A grain storage facility consists of a right circular cylindrical silo of height h meters and radius r meters, topped by a hemispherical roof of the same radius. The volume of grain to be stored is 4000 m3. The curved surface of the cylindrical wall costs \120persquaremetertoconstruct,whilethecurvedsurfaceofthehemisphericalroofcosts180 per square meter. No cost is associated with the silo floor, and the roof and wall materials can be purchased in any amount.
(a) Write an expression for the total construction cost K (in dollars) in terms of r and h. Clearly label the contributions of the wall and the roof.
(b) Use the volume requirement to rewrite K as a function of a single variable.
(c) Determine the values of r and h that minimize K. Provide exact values or values correct to three significant figures, and justify that the critical point found corresponds to the absolute minimum cost.
(d) Discuss how the optimal radius would change if the roof material became less expensive than the wall material while the total required volume remained fixed. Support your answer with derivative-based reasoning.