## Brain-Teaser 470

**From The Sunday Times, 31st May 1970** [link]

Joseph’s garage, built at ground level on to the side wall of his house, had uniform external dimensions of (exactly) 10 ft. high by 10 ft. wide.

Wishing to paint from the outside a top-floor window-frame over the garage, he positioned his longest ladder (whose length was over 30 ft.) with its foot on the ground at such a distance from the garage that its top rested against the house wall at the maximum height possible.

Discovering, however, that despite this he could not reach up to the window, he borrowed his neighbour Jacob’s ladder and, positioning it similarly, found that he could now reach comfortably.

With either ladder the length, distance and height referred to were all integral numbers of inches.

How much longer was Jacob’s ladder than Joseph’s, and how much higher on the wall did it reach?

[teaser470]

## Jim Randell 8:02 am

on11 April 2019 Permalink |For a ladder of length

k, the maximum height we could achieve would bek, if we laid it flat against the wall, but we can’t do that because there is a garage in the way. If we position it at some distance away from the garage, so that it reaches over the garage we find that as we move the foot of the ladder towards the garage the other end will reach higher up the wall, until the garage gets in our way.If the foot of the ladder is at a distance

dfrom the wall, and a heighthup the wall, and touches the garage at a point 120 inches from the wall and 120 inches above the ground, then:and the length of the ladder

kis given by:This program increases

d(in inches from 120) looking for integer heights with integer length ladders.Run:[ @repl.it ]Solution:Jacob’s ladder was 4 ft. 3 in. longer than Joseph’s, and reached 5 ft. 3 in. higher.In the diagram:

Jacob’s ladder (blue) is 442 inches long (= 36 ft. 10 in.), and reaches a height of 408 inches (= 34 ft.) on the wall.

Joseph’s ladder (red) is 391 inches long (= 32 ft. 7 in.) and reaches a height of 345 inches (= 28 ft. 9 in.) on the wall.

There is a further possible ladder (green) which is 350 inches (= 29 ft. 2 in.) and would reach a height of 280 inches (= 23 ft. 4 in.) on the wall, but this ladder is not longer than 30 ft.

We can also swap the distance and height to position the ladders, but this results in a maximum distance rather than a maximum height achieved.

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