Problem 31 - Essential Calculus

Question

A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)

Step-by-Step Solution

Answer

The maximum volume that can be sent is with a package of dimensions 9 inches by 9 inches by 72 inches.

Step 1: Setup the function and constraints

Given that the package cross section is a square, let each side of the square be \(x\) and the length of the box be \(y\). The girth of the package is the perimeter of this square, so it is \(4x\). Thus, we have \(4x + y = 108\) as our constraint equation. For a right rectangular box, the volume \(V\) is given by \(V = x^2 * y\). Thus, our objective is to maximize this volume.

Step 2: Substitute the constraint into the volume function

To find a solution, it is necessary to have the volume function in terms of one variable. This can be achieved by rearranging the constraint equation for \(y\) and substituting into the volume function. From the constraint \(4x + y = 108\), we have \(y = 108 - 4x\). Substituting this into the volume function gives 👉 \(V = x^2 * (108 - 4x)\).

Step 3: Find the critical points

To find the maximum of the volume function, we will take derivatives. The first derivative of \(V\) is \(V' = 108x - 12x^2\). Set this equal to zero and solve for \(x\), we have \(108x - 12x^2 = 0\). Solving this equation gives \(x = 0\) and \(x = 9\). Since \(x = 0\) is not a practical solution (it would imply a package of zero girth), so the side length that maximizes package volume is \(x=9\) inches.

Step 4: Find the corresponding value of y

Substitute \(x = 9\) into the constraint equation to find the associated value of \(y\). From \(4x + y = 108\), substituting \(x=9\) we get \(y = 108 - 36 = 72\) inches. Therefore, the dimensions of the package of maximum volume are a square cross-section of side length 9 inches and a length of 72 inches.