A grain storage facility consists of a right circular cylindrical silo of height and…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A grain storage facility consists of a right circular cylindrical silo of height hm and radius rm, topped by a hemispherical roof of the same radius. The volume of grain to be stored is 4000m3. The curved surface of the cylindrical wall costs $120 per square metre to construct, while the curved surface of the hemispherical roof costs $180 per square metre. 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 minimise 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.