Expert: Ahmed Salami - 12/12/2008

Question
A private shipping company will accept a box for domestic shipment if the sum of its length and girth (distance around) does not exceed 90 inches. Suppose you want to mail a box with square sides so that its dimensions are h by h by w and its girth is 2h + 2w. What dimensions will give the box the largest volume?

Answer
Hi Josh,
For the described box,
V = h.h.w
 = wh^2
but,
2h + 2w = 90
h + w = 45
w = 45 - h
So,
V = (45-h)h^2
 = 45h^2 - h^3
dV/dh = 90h - 3h^2
The largest volume occurs when dV/dh = 0, hence
90h - 3h^2 = 0
3h(30 - h) = 0
i.e at h = 0 or h = 30
We obviously discard the first value and take h = 30. Or we could be more mathematic. The maximum volume occurs when d^2V/dh^2 (the second differential) is negative at the point where dV/dh is zero.
d^2V/dh^2 = 90 - 6h
at h = 0, d^2V/dh^2 = 90 (positive)
at h = 30, d^2V/dh^2 = 90 - 180 = -90 (negative)
And so our required value is h = 30
w = 45 - 30
 = 15

Regards