Using Local Extrema to Solve Applications

We will start this problem by drawing a picture like that in Figure 22, labeling the width of the cut-out squares with a variable, $w.$

Notice that after a square is cut out from each end, it leaves a $\left(14-2w\right)$ cm by $\left(20-2w\right)$ cm rectangle for the base of the box, and the box will be $w$ cm tall. This gives the volume

Notice, since the factors are $w,$ $20–2w$ and $14–2w,$ the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider the only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values $w$ may take on are greater than zero or less than 7. This means we will restrict the domain of this function to $0<w<7.$ Using technology to sketch the graph of $V\left(w\right)$ on this reasonable domain, we get a graph like that in Figure 23. We can use this graph to estimate the maximum value for the volume, restricted to values for $w$ that are reasonable for this problem—values from 0 to 7.

From this graph, we turn our focus to only the portion on the reasonable domain, $\left[0,\text{<mn>7</mn>}\right].$ We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce Figure 24.

From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.