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Calculus

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The Volume of a box with a square base x by x cm and height h cm is:

$V=x^{2} h$

We know V=70304 cm3

$h=\frac{70304}{x^{2} } $

So, the area is:

$A=x^{2} +4x(\frac{70304}{x^{2} } )=x^{2} +\frac{281216}{x} $

We want to minimize surface area (A), so we calculate the differential:

$A^{'} =2x-\frac{2881216}{x^{2} } =0$

So, we have:

$x=52$

And so we have:

$h=\frac{70304}{x^{2} } =26$

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