Problem:
A straight river that is meters wide flows from west to east at a rate of meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of meters downstream from Sherry. Relative to the water, Melanie swims at meters per minute, and Sherry swims at meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find .
Solution:
Answer (550):
Because the two women cross the river in the same amount of time, the north-south components of their velocities are the same value . The east-west components of Melanie's and Sherry's velocities must be values and , respectively, because the two women meet halfway between their starting points. But Melanie is swimming against the river's current, and Sherry is swimming with the river's current, so relative to the water, Melanie's velocity is given by the vector , and Sherry's velocity is given by the vector . Thus the squares of their speeds are
Subtracting and solving for yields , from which . It takes each woman minutes to complete her swim. Each woman swims along the river a distance of meters, so .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions