## You are here

Homecuboid with least surface

## Primary tabs

# cuboid with least surface

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

$\displaystyle\begin{cases}{f^{{\prime}}_{x}(x,\,y)=y-\frac{k^{3}}{x^{2}}=0},\\ {f^{{\prime}}_{y}(x,\,y)=x-\frac{k^{3}}{y^{2}}=0}.\end{cases}$ |

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 $f^{{\prime}}_{x}$ and $f^{{\prime}}_{y}$ both vanish.

## Mathematics Subject Classification

26B12*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections