Henry Ernest Dudeney/Modern Puzzles/142 - Economy in String/General Solution

by : $142$

 * Economy in String

General Result
Let the string pass:
 * $a$ times along length $x$
 * $b$ times along breadth $y$
 * $c$ times along depth $z$.

Let the string be length $m$.

Then the maximum volume $xyz$ of the parcel is given by:


 * $x y z = \dfrac {m^2} {27 a b c}$

where:

Proof
We have that:
 * $a x + b y + c z = m$

The maximum area of $x y$ is found as follows:

Put:

Similarly also by differentiation $x$:
 * $a x = \dfrac n 2$

and so:
 * $a x = b y$

Similarly:
 * $a x = b y = c z = \dfrac m 3$

and so:

Hence:


 * $x y z = \dfrac {m^2} {27 a b c}$