11. Two Ferryboats

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TWO FERRYBOATS start at the same instant from opposite sides of a river,
traveling across the water on routes at right angles to the shores. Each travels at a
constant speed, but one is faster than the other. They pass at a point 720 yards
from the nearest shore. Both boats remain in their slips for 10 minutes before
starting back. On the return trips they meet 400 yards from the other shore.
How wide is the river?

The answer

When the ferryboats meet for the first time [top illustration], the combined
distance traveled by the boats is equal to the width of the river. When they reach
the opposite shore, the combined distance is twice the width of the river; and
when they meet the second time [bottom figure], the total distance is three times
the river’s width. Since the boats have been moving at a constant speed for the
same period of time, it follows that each boat has gone three times as far as when
they first met and had traveled a combined distance of one river-width. Since the
white boat had traveled 720 yards when the first meeting occurred, its total
distance at the time of the second meeting must be 3 X 720, or 2,160 yards. The
bottom illustration shows clearly that this distance is 400 yards more than the
river’s width, so we subtract 400 from 2,160 to obtain 1,760 yards, or one mile,
as the width of the river. The time the boats remained at their landings does not
enter into the problem.
The problem can be approached in other ways. Many readers solved it as
follows. Let x equal the river-width. On the first trip the ratio of distances
traveled by the two boats is x – 720:720. On the second trip it is 2x – 400 😡 +
400. These ratios are equal, so it is easy to solve for x.

 

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