The dimensions of an acceptable box is 18 x 36 inches.
If the package has a combined length and girth of 108 inches. We can determine that:
4x + y = 108
We are also assuming that the cross section is square, so:
V = x²y
Let's solve for y in the first equation and plug it in to the second equation:
y = 108- 4x
V = x²(108- 4x)
V = 108x² - 4x³
In order to solve for the maximum is when V' = 0
V' =216x - 12x² = 0
x(216- 12x) = 0
Therefore, we have 2 solutions,
x = 0
216- 12x = 0
12x = 216
x = 18
x = 0 is when the package is a minimum, so the maximum occurs when x = 18. Now let's solve for y
y = 108- 4x = 108- 4(18) = 36
The answers are:
x = 18 inches
y = 36 inches
Therefore, the dimensions of an acceptable box is 18 x 36 inches.
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