Q. 47 - Precalculus Enhanced with Graphing Utilities

Question

(a) Express the surface area S of the box as a function of x.

(b) Using a graphing utility, graph the function found in

part (a).

(c) What is the minimum amount of cardboard that can be

used to construct the box?

(d) What are the dimensions of the box that minimize the

surface area?

(e) Why might UPS be interested in designing a box that

minimizes the surface area?

Step-by-Step Solution

Answer

Surface area is minimized when x = cube root of 2V.

Step 1: Surface Area Function

For a box with square base \(x\) and fixed volume \(V\), height \(h = V/x^2\).
\(S(x) = x^2 + 4xh = x^2 + 4V/x\)

Step 2: Minimize

\(S'(x) = 2x - 4V/x^2 = 0 \Rightarrow x = \sqrt[3]{2V}\). The minimum surface area and optimal dimensions depend on the specific volume.

Step 3: Why UPS Cares

Minimizing surface area reduces material costs for box manufacturing, saving money and resources.