Henry Ernest Dudeney/Modern Puzzles/45 - The Man and the Dog/Solution

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Modern Puzzles by Henry Ernest Dudeney: $45$

The Man and the Dog
"Yes, when I take my dog for a walk," said a mathematical friend, "he frequently supplies me with some interesting problem to solve.
One day, for example, he waited, as I left the door, to see which way I should go,
and when I started he raced to the end of the road, immediately returning to me;
again racing to the end of the road and again returning.
He did this four times in all, at a uniform speed,
then ran at my side the remaining distance, which according to my paces measured $27$ yards.
I afterwards measured the distance from my door to the end of the road and found it to be $625$ feet.
Now, if I walk $4$ miles per hour, what is the speed of my dog when racing to and fro?"


Solution

The speed of the dog is $16$ miles per hour.


Proof

Recall that $1$ mile is $1760$ yards, while $1$ yard is $3$ feet.

Therefore the dog owner is walking at a pace of $21 \, 120$ feet per hour, with $81$ feet remaining after the four trips.


Suppose the dog is running at a pace of $x$ feet per hour.

Suppose after some trips, the owner still $d$ feet of the road to go.

For the next trip, the owner and the dog would have travelled a total of $2 d$ feet.

With their combined speed of $\paren {x + 21 \, 120}$ feet per hour, this trip would have taken $\dfrac {2 d} {x + 21 \, 120}$ hours.

The owner would have walked an additional $\dfrac {21 \, 120 \times 2 d} {x + 21 \, 120}$ feet, so there would be

$d - \dfrac {21 \, 120 \times 2d } {x + 21 \, 120} = \dfrac {d \paren {x - 21 \, 120} } {x + 21120}$

feet of road to cover, $\dfrac {x - 21 \, 120} {x + 21 \, 120}$ of the previous distance.


Thus after four trips, the remaining distance is $\paren {\dfrac {x - 21120} {x + 21120} }^4$ of the original.

We have:

\(\ds 625 \paren {\frac {x - 21 \, 120} {x + 21 \, 120} }^4\) \(=\) \(\ds 81\)
\(\ds \paren {\frac {x - 21\, 120} {x + 21 \, 120} }^4\) \(=\) \(\ds \frac {3^4} {5^4}\)
\(\ds 5 \paren {x - 21 \, 120}\) \(=\) \(\ds 3 \paren {x + 21 \, 120}\)
\(\ds x\) \(=\) \(\ds 84 \, 480\)

Hence the speed of the dog is $84 \, 480$ feet per hour, or $16$ miles per hour.

$\blacksquare$


Sources

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