Maximize the volume of a lidless box formed from 5x8 sheet with a square (with a length of x) removed from each corner.
Using the figure above, the dimensions of the box are
The formula for the volume of this box would be
Use a spreadsheet to display possible values of x, (8-2x), (5-2x), and V.
We only need to use values of x<2.5 since the width is 5 and using a larger value of x would give us a negative number and the sheet would disappear.
Looking at the Volume column shows that the volume appears to reach a maximum volume of 18 when x=1.
This is supported using the first derivative of V.
The first derivative of V gives the critical points of 1 and 10/3. The value of 1 is the only value between 0 and 2.5.
Using the second derivative of V,
the value is positive when x=1 and that means V is a maximum at that value.
This can also be verified by the graph of the equation
At x=1, the graph shows a relative maximum point.
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