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

by : $45$

 * The Man and the Dog

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:

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