An aquarium designer plans to build an open-top rectangular tank that must hold…
MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
An aquarium designer plans to build an open-top rectangular tank that must hold exactly 600 liters of water (1 liter = 1,000 cm³). The base of the tank will be constructed from a tempered glass panel that costs twice as much per square centimeter as the glass used for the side panels.
(a) Derive an expression for the total material cost C (in arbitrary cost units) in terms of the length ℓ, width w, and height h of the tank.
(b) Using the fixed-volume requirement, reduce the cost expression so that it depends on no more than two of the three dimensions.
(c) Explain how a second substitution can be used to express C as a function of a single variable, and justify the appropriate domain for that variable.
(d) Describe, without carrying out the differentiation, the calculus-based procedure for locating the minimum possible cost and verifying that the critical point within the domain corresponds to that minimum.