You would like to constructbox with open top & square base with a volume of 8 cubic feet.?
What are the dimensions of the box that uses the least amount of material to construct? Be sure to explain, via the first or second derivative test, why we really do have a minimum.
V = s^2 * h, where s is the side of the square base, and h is the height.
in terms of s, h = V / s^2
we know that V = 8, so h = 8/s^2
Since this is is an open top, the surface area will be :
A = s^2 + 4s * h
but h = 8/s^2, so now we have:
A(s) = s^2 + 4s*8/s^2 = s^2 + 32/s
This is the function we’re trying to minimize
A ‘ = 2s – 32/s^2
set equal to 0
0 = 2s – 32/s^2
2s = 32/s^2
s^3 = 16
s = 16^1/3 = 2 * cbrt(2)
h = 8/s^2 = 8/(4*2^(2/3)) = 2 / [2^(2/3)] = cbrt(2)
so the box will be 2cbrt(2) x 2cbrt(2) x cbrt(2) (V = 8 check)
Area used will be 4 * 2^(2/3) + 8 * 2^(2/3) = 12 * 2^(2/3)
check: A” = 2 + 64/s^3, which is > 0 for positive s, and therefore gives a minimum for A
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