Henry Ernest Dudeney/Modern Puzzles/142 - Economy in String/Solution/Proof 2
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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?
Solution
$2$ feet by $1$ foot by $\tfrac 2 3$ feet.
Proof
From the general solution:
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:
\(\ds x\) | \(=\) | \(\ds \dfrac m {3 a}\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \dfrac m {3 b}\) | ||||||||||||
\(\ds z\) | \(=\) | \(\ds \dfrac m {3 c}\) |
$\Box$
Setting:
\(\ds a\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds 4\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds m\) | \(=\) | \(\ds 12\) |
we obtain:
\(\ds x\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds z\) | \(=\) | \(\ds \dfrac 2 3\) | ||||||||||||
\(\ds x y z\) | \(=\) | \(\ds 1 \tfrac 1 3\) |
Hence the result.
$\blacksquare$
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