(Originais Teachy 2023) - Question Hard of Mathematics

A revolution cylinder is a solid whose lateral faces are two identical and parallel cylindrical surfaces and whose bases are two identical and parallel circular discs. An architecture student is designing a cylindrical water tower for a water reuse project in a community. The tower must have a water storage capacity of 5000 cubic meters. Additionally, the student wants to externally coat the tower with a material that costs R$ 10.00 per square meter. Considering that the tower should be as efficient as possible in terms of material usage for coating, the student wants to know what should be the height and radius of the tower to minimize the coating cost, knowing that the volume of a cylinder is given by V = π * r^2 * h and the surface area is given by A = 2 * π * r * (r + h). Based on these calculations and the cost function C = 10 * A, where C is the cost of the coating, determine the height and radius that minimize the coating cost.