|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
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
 |
(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
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.
|
"cuboid with least surface" is owned by pahio.
|
|
(view preamble | get metadata)
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
There is 1 reference to this entry.
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 1116 times total.
Classification:
| AMS MSC: | 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![\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)}$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/9770/js/img3.png "\begin{pspicture}(-0.5,-0.5)(3.5,4) \psaxes[Dx=9,Dy=9]{->}(0,0)(-0.5,-0.5)(5.5,4... ...y = M$} \psdot[linecolor=red](1,1) \rput(1.4,1.15){$^{(k,\,k)}$} \end{pspicture}")