Henry Ernest Dudeney/Puzzles and Curious Problems/239 - The Bell Rope/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $239$

The Bell Rope
A bell rope, passing through the ceiling above, just touches the belfry floor,
and when you pull the rope to the wall, keeping the rope taut, it touches a point just $3$ inches above the floor,
and the wall was $4$ feet from the rope, when it hung at rest.
How long was the rope from floor to ceiling?


Solution

$32$ feet and $1 \tfrac 1 2$ inches.


Proof

Let $L$ inches be the length of the bell-rope.

Thus we have described a right triangle:

whose legs are $4 \times 12 = 48$ and $L - 3$ (as there are $12$ inches to the foot)
and whose hypotenuse is $L$.

Thus:

\(\ds \paren {L - 3}^2 + 48^2\) \(=\) \(\ds L^2\) Pythagoras's Theorem
\(\ds \leadsto \ \ \) \(\ds L^2 - 6 L + 9 + 2304\) \(=\) \(\ds L^2\) multiplying out
\(\ds \leadsto \ \ \) \(\ds L\) \(=\) \(\ds \dfrac {2313} 6\) simpliication
\(\ds \) \(=\) \(\ds 385 \tfrac 1 2\) simpliication

The result follows on converting $385 \tfrac 1 2$ inches to feet and inches.

$\blacksquare$


Sources

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