A packaging company will produce an open-top rectangular box with a square base that…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A packaging company will produce an open-top rectangular box with a square base that must hold exactly 500 cm3 of dry product. The material used for the base costs twice as much per square centimeter as the material used for the four sides. Ignore any cost associated with seams.
Let x denote the length (in cm) of one side of the square base, and let h denote the height (in cm) of the box.
(a) Express the total construction cost C (in cost units) in terms of x and h. Clearly indicate which portions of the box correspond to which terms in your expression.
(b) Using the volume requirement, rewrite C as a function of a single variable.
(c) Determine the dimensions x and h that minimize C. Show all analytical work leading to the identification of critical points and classify the nature of the extremum.
(d) Interpret your result in context. Include units in your discussion and state the minimum possible construction cost to the nearest tenth of one cost unit.