UNO High School Problem of the Week Competition
Problem 5
Two men at points R and S, 76 miles apart, set out at the same time to walk towards each other. The man at R walks uniformly at the rate of 4 1/2 miles per hour; the man at S walks at the constant rate of 3 1/4 miles per hour for the first hour, at 3 3/4 miles per hour for the second hour, and so on, in arithmetic progression.
The men will meet x miles nearer R than S, find x.
Solution
Let h = number of hours in which they meet. We may then come up with the equation
9/2h + 1/2h[2 * 13/4 + (h-1)1/2 ] = 76
Thus, we get the quadratic,
h^2 + 30h -304 = 0
Factoring we can arrive at
(h - 8)(h + 38) = 0
Giving us h = 8 or -38. Since we can't have a negative time, h = -38, is an irrational answer. Thus after 8 hours the two men will meet.
The meeting point is then 36 miles from R and 40 miles from S. We are 4 miles closer to R than S, so x = 4.
