Exercise 39

The airlines place restrictions on luggage that can be carried onto planes.

• A carry-on bag can weigh no more than 40 lbs.

• The length plus width plus height of a bag cannot exceed 45 inches.

• The bag must fit in an overhead bin.

Let $x$, $y$, and $z$ be the length, width, and height (in inches) of a carry on bag. In this problem we find the dimensions of the bag of largest volume, $V = xyz$, that satisfies the second restriction. Assume that we use all 45 inches to get a maximum volume. (Note that this bag of maximum volume might not satisfy the third restriction.)

1. Write the volume $V=V(x,y)$ as a function of just the two variables $x$ and $y$.

2. Explain why the domain over which $V$ is defined is the triangular region $R$ with vertices (0,0), (45,0), and (0,45).

3. Find the critical points, if any, of $V$ in the interior of the region $R$.

4. Find the maximum value of $V$ on the boundary of the region $R$, and then determine the dimensions of a bag with maximum volume on the entire region $R$. (Note that most carry-on bags sold today measure $22$ by $14$ by $9$ inches with a volume of $2772$ cubic inches, so that the bags will fit into the overhead bins.)