Two boats on opposite sides of a river head towards each other at different speeds. When they pass each other the first time they are 700 yards from one shoreline. They continue to the opposite shoreline, turn around, and move towards each other again. When they pass the second time they are 300 yards from the other shoreline.

How wide is the river? (Assume both boats travel at a constant speed and ignore factors such as turn-around time and the current of the river).

Let X be the width of the river in yards. The boat coming from the east bank is A, from the west bank is B. The first rendevous point is 700 miles from the east bank. The second is 300 miles from the west bank.

In time interval t1, A sails 700 yards west and B sails
X-700 yards east. In time interval t2, A sails X-700 yards west, then X-300 yards east, for a total of 2X-1000 yards, while B sails 700 yards east and 300 yards west, for a total of 1000.

Since the two boats' relative speeds remain constant, the ratio of the distances they sail in t1 should be equal to the ratio of the distances they sail in t2.

700/(X-700) = (2X-1000)/1000
350/(X-700) = (X-500)/1000
350000 = (X-700)*(X-500)
X^2 - 700X - 500X + 350000 = 350000
X^2 -1200X = 0
X(X-1200) = 0
X=1200

The river is 1200 yards wide.

Edited on November 13, 2003, 9:04 pm

Posted by Dan on 2003-11-13 20:07:08