A manufacturer needs to design a closed rectangular shipping box with a square base…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A manufacturer needs to design a closed rectangular shipping box with a square base that will hold exactly 4,000 cm3 of product. The company wishes to minimize the amount of cardboard used for the box.
(a) Let x be the length of a side of the square base (in cm) and h be the height of the box (in cm). Write an equation that relates x, h, and the required volume.
(b) Using the same variables, write a formula for the total exterior surface area S (in cm2) of the box, including all six faces.
(c) Eliminate one variable to rewrite the surface‐area function S solely in terms of x. Clearly state the domain of this single‐variable function within the physical context.
(d) Determine the value of x that minimizes the surface area. Then compute the corresponding height h and the minimum surface area to the nearest whole square centimeter.
(e) Interpret your results in the context of material usage, explaining why the dimensions you found indeed minimize the amount of cardboard required.