how to calculate surface area of rectangular prism with given volume

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I am doing a math project on optimisation and have a rectangular prism with a volume of 100cm3 but no other information. I am supposed to analyse whether the manufacturer of that product has designed the optimal package to hold that volume. Any help?

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2 Answers

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Volume:

$$ V=LWH=100 $$

Surface area:

$$ S=2LW+2LH+2WH $$

Using Lagrange multipliers:

$$ \nabla S=\lambda\nabla V $$

$$ \nabla S =(2W+2H,2L+2H,2L+2W) $$

$$ \nabla V=(WH,LH,LW) $$

\begin{eqnarray} 2W+2H&=&WH\lambda\\ 2L+2H&=&LH\lambda\\ 2L+2W&=&LW\lambda \end{eqnarray}

Resulting in

\begin{eqnarray} \lambda&=&\frac{2}{H}+\frac{2}{W}\\ &=&\frac{2}{H}+\frac{2}{L}\\ &=&\frac{2}{W}+\frac{2}{L} \end{eqnarray}

So $L=W=H$. Therefore $LWH=100$ gives $L^3=100$, so $L=W=H=10^{2/3}$.

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We know that $wlh=v$ and we can use this relation to eliminate $h$.

Then the (half) area is

$$wl+lh+hw=wl+\frac vw+\frac vl.$$

We minimize it by canceling the gradient,

$$\begin{cases}l-\dfrac v{w^2}=0,\\w-\dfrac v{l^2}=0\end{cases},$$ obviously giving $l=w=h$.

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