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).

The width of the river is (700+x+300) yards wide.

After 1*T, the two boats meet for the first time.
Boat_a travelled 700 yards.
Boat_b travelled 300+x yards.

Each boat is given to travel at a constant rate.
Boat_a travels at a rate of 700/T.
Boat_b travels at a rate of 300+x/T.

After n*T, the two boats meet for the second time.
Boat_a travelled ((700+x+300)+(300)) = 1300+x yards.
Boat_b travelled ((300+x+700)+(700+x)) = 1700+2x yards.

The equation for Boat_a in regard to distance travelled at the given rate of speed when the boats meet the second time can be expressed as: 1300+x = 700*n. n, therefore, is (1300+x)/700.

The equation for Boat_b in regard to distance travelled at the given rate of speed when the boats meet the second time can be expressed as: 1700+2x = (300+x)*n.

By substitution we have
1700+2x = (300+x)(1300+x)/700.

Simplifying into a quadratic equation:
x² + 200x - 800000 = 0

Applying the quadratic formula, (discarding the negative value, distance being positive), we have x = 800.

The width of the river is, then, (700+800+300) = 1800 yards wide.

Posted by Dej Mar on 2008-07-16 00:59:06