A box with an open top has vertical sides, a square bottom, and a volume of 256 cubic meters. If the box has the least possible surface area, find its dimensions.

Question

aaronq · Answer

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let h be the height and x the side lengths

$V=h*x^2=256~m^3\rightarrow h=\dfrac{256~m}{x^2}$

$SA=x^2+4hx$    If we only count the outside faces.
$SA=2x^2+8hx$   If we count the inside faces as well as the outside faces. 

Well go with the second one, sub in h

$SA=2x^2+8(\dfrac{256}{x^2})x=2x^2+\dfrac{2048}{x}$  

Now, we take the first derivative and let it equal to zero.

$SA'=4x-\dfrac{2048}{x^2}=0$  

Solve for x and find h with the initial equation.