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A postal service will accept packages only if the length plus girth is no more than...

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A postal service will accept packages only if the length plus girth is no more than 220 inches. (See the figure.) Assuming that the front face of the package (as shown in the figure) is square, what is the largest volume package that the postal service will accept?

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Answer #1
We will call the length of the side of a face s and the length of the package l:

4s + l = 220 (length + girth)

Now we want to find the volume and find the maximum:

V = s^2*l

We want to use the length + girth equation to substitute for l:

l = 220 - 4s

V = s^2*(220 - 4s) = 220s^2 - 4s^3

We now take the derivative and set it equal to 0 to find the maximum:

dV/ds = 440s - 12s^2 = 0

440 - 12s = 0

12s = 440
s = 36.67 inches

Now we plug this into the volume equation to find the maximum volume:

V = 220s^2 - 4s^3 = 98592.59 in^3
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