Henry Ernest Dudeney/Modern Puzzles/142 - Economy in String/General Solution/Proof 1

by : $142$

 * Economy in String

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:


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