An open cylindrical storage tank must hold 500 liters of liquid (1 liter = 1000 cm )
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
An open cylindrical storage tank must hold 500 liters of liquid (1 liter = 1000 cm3). The tank is to be constructed from sheet metal costing 0.04percm^{2}forthecircularbaseand0.02 per cm2 for the curved side. For transportation reasons, the diameter of the tank may not exceed 120 cm.
Let r cm be the radius of the base and let h cm be the height of the tank.
(a) Write the total construction cost C of the tank in terms of r and h. Clearly separate the cost contributions of the base and the side.
(b) Use the volume requirement to express C as a function of r only and state the feasible interval for r imposed by the transportation constraint and physical considerations.
(c) Analyze the function from part (b) to locate all interior critical points and evaluate the total cost at those points.
(d) Evaluate the total cost at any endpoint(s) of the feasible interval identified in part (b).
(e) Determine the radius, height, and total cost corresponding to the least expensive feasible tank, and justify that your choice yields the absolute minimum cost under all given constraints.