how to calculate surface area of rectangular prism with given volume
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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$\begingroup$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 $L=W=H=10^{2/3}$.
$\endgroup$ 1 $\begingroup$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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