A closed right circular cylinder is required to contain exactly 1,000 cm3 of fluid. Let r cm be the radius and h cm be the height of the cylinder.
(a) Write an expression for the total exterior surface area A(r,h) of the cylinder, including both bases.
(b) Use the fixed-volume condition to express h in terms of r, and write A as a function A(r) of one variable.
(c) Determine the dimensions r and h that minimize the total exterior surface area.
(d) Calculate the minimum surface area to the nearest whole square centimeter and interpret what this value represents in the manufacturing context.