An architect is designing a closed cylindrical water tower that must have a fixed…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
An architect is designing a closed cylindrical water tower that must have a fixed volume of 200m3. The material for the circular top and bottom costs twice as much per square meter as the material for the lateral surface.
Let r be the radius (in meters) of the cylinder’s base, and let h be its height (in meters). Assume all waste is negligible.
(a) Derive an expression for the total construction cost C(r) as a function of r alone. Clearly indicate any constants of proportionality you introduce and state the domain of r that is physically meaningful.
(b) Determine the radius and height that minimize the total cost, expressing your answers exactly.
(c) Verify that the critical point you obtained yields a global minimum on the domain of interest, citing appropriate calculus reasoning.
(d) Discuss how the optimal dimensions would change, if at all, if the material for the top and bottom were instead three times as expensive as the lateral material.