A cylindrical storage tank is designed to hold 10m3 of liquid at its interior capacity. After fabrication, the tank is completely wrapped (sides and top) with a uniform insulating layer of thickness t=0.05m. The bottom of the tank is not insulated. Let r and h denote the interior radius and interior height (both in meters) of the tank before insulation is applied.
(a) Derive an expression, in terms of r, h, and the fixed thickness t, for the volume Vins of insulating material that will be used.
(b) Use the requirement that the interior volume is fixed at 10m3 to eliminate h from your expression and obtain Vins as a function of r alone.
(c) Determine the interior radius that minimizes the amount of insulation material required. Give your answer to three decimal places.
(d) Confirm that your result in part (c) corresponds to a minimum amount of insulation.
(e) Explain, without further calculation, how increasing the insulation thickness t would qualitatively affect the optimal interior radius and height.