Maximize the volume of the lidless box formed from a 5 x 8 sheet with a square removed from each corner.
I used EXCEL to make a spreadsheet to do all the calculations and draw a graph. The first column represents the x which is the amount cut from one corner.
The second column is the length, calculated by subtracting 2x from 8. Similarly, the width is calculated by subtracting 2x from 5 and the height is the value of x.
Click here to see the GSP sketch. You can drag the "Big Guy" point to change the value of x.
As you can see, the volume is maximized at x = 1, l = 6, w = 3, and h = 1. The volume is 18. Continuing the spreadsheet down, the values will get negative because you begin taking out more of a side than is actually there.
Looking at the picture we see another way of looking at the maximum.
The curve shows us once again that the maximum volume is when x = 1 which is where the curve reaches its maximum.