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[parent] cuboid with least surface (Example)

Let us determine among all cuboids (i.e. rectangular parallelepipeds) with a given volume $ k^3$ such one which has the least surface area.

Let the three edges of the cuboid beginning from a vertex be $ x$, $ y$ and $ z$; then we must start from the condition $ xyz = k^3$, whence $ z = \frac{k^3}{xy}$. We get the expression

$\displaystyle f(x,\,y) := 2(yz\!+\!zx\!+\!xy) = 2\!\left(\!xy+\frac{k^3}{x}+\frac{k^3}{y}\!\right)$ (1)

for the whole area of the surface of the cuboid. Thus we have to make $ f(x,\,y)$ a minimum, when only the positive values of $ x$ and $ y$ can be taken into consideration.

The function $ f$ and its first order partial derivatives are continuous for all positive $ x$ and $ y$. According to the theorem of the parent entry, a minimum can occur only when simultaneously

\begin{align*}\begin{cases}{f'_x(x,\,y) = y-\frac{k^3}{x^2} = 0},\\ {f'_y(x,\,y) = x-\frac{k^3}{y^2} = 0}. \end{cases}\end{align*}    

These equations are true only for $ x = y = k$, i.e. for the case that the cuboid is a cube. We can infer that a cube has the least area. In fact, we see from (1) that $ f(x,\,y)\to\infty$ as $ x\to 0$ or $ y\to 0$ or $ xy\to\infty$; therefore there exist a small positive number $ m$ and a big positive number $ M$ such that outside and on the boundary of the region resembling a triangle and bounded by the lines $ x = m$ and $ y = m$ and the rectangular hyperbola $ xy = M$, the value of $ f(x,\,y)$ is always greater than in the point $ (k,\,k)$ inside this region. Thus the function gets its smallest value in an interior point of the region, and this point must be $ (k,\,k)$ since it is the only point where $ f'_x$ and $ f'_y$ both vanish.

\begin{pspicture}(-0.5,-0.5)(3.5,4) \psaxes[Dx=9,Dy=9]{->}(0,0)(-0.5,-0.5)(5.5,4... ...y = M$} \psdot[linecolor=red](1,1) \rput(1.4,1.15){$^{(k,\,k)}$} \end{pspicture}



"cuboid with least surface" is owned by pahio.
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Cross-references: vanish, interior point, point, rectangular hyperbola, lines, bounded, triangle, region, boundary, number, cube, equations, continuous, partial derivatives, first order, function, positive, area, expression, vertex, edges, surface area, volume, parallelepipeds
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This is version 4 of cuboid with least surface, born on 2007-07-15, modified 2007-09-04.
Object id is 9770, canonical name is CuboidWithLeastSurface.
Accessed 580 times total.

Classification:
AMS MSC: 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)
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