Henry Ernest Dudeney/Modern Puzzles/142 - Economy in String/General Solution/Proof 1
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Modern Puzzles by Henry Ernest Dudeney: $142$
- Economy in String
- Owing to the scarcity of string a lady found herself in this dilemma.
- In making up a parcel for her son, she was limited to using $12$ feet of string, exclusive of knots,
- which passed round the parcel once lengthways and twice round its girth, as shown in the illustration.
- What was the largest rectangular parcel that she could make up, subject to these conditions?
General Result
Let the string pass:
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:
\(\ds x\) | \(=\) | \(\ds \dfrac m {3 a}\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \dfrac m {3 b}\) | ||||||||||||
\(\ds z\) | \(=\) | \(\ds \dfrac m {3 c}\) |
Proof
We have that:
- $a x + b y + c z = m$
The maximum area of $x y$ is found as follows:
Put:
\(\ds a x + b y\) | \(=\) | \(\ds n\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \dfrac {n - b y} a\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x y\) | \(=\) | \(\ds \dfrac {n y} a - \dfrac {b y^2} a\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac \d {\d y} x y\) | \(=\) | \(\ds \dfrac n a - \dfrac {2 b y} a\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \dfrac n {2 b}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds b y\) | \(=\) | \(\ds \dfrac n 2\) |
Similarly also by differentiation with respect to $x$:
- $a x = \dfrac n 2$
and so:
- $a x = b y$
Similarly:
- $a x = b y = c z = \dfrac m 3$
and so:
\(\ds x\) | \(=\) | \(\ds \dfrac m {3 a}\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \dfrac m {3 b}\) | ||||||||||||
\(\ds z\) | \(=\) | \(\ds \dfrac m {3 c}\) |
Hence:
- $x y z = \dfrac {m^2} {27 a b c}$
$\blacksquare$
Historical Note
According to Dudeney, this general solution is the result of work by Alexander Yule Fraser.
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $142$. -- Economy in String
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $313$. Economy in String