Find the Speed
This problem is from Robert Stanfill by way of Tim Coburn.
Outdoorsman Russ Rustic was rowing upstream on a river at a constant rate, thoroughly enjoying the scenic wonder of the area. In fact, he was so preoccupied with the beauty that he did not notice that his hat blew off and landed in the river just as he was rowing under a bridge. Exactly five minutes later he discovered his loss and immediately turned around and started rowing downstream with the same level of vigor that he was using while rowing upstream, hoping to find his lost hat.
Even though his hat floated downstream at the same rate as the river the entire time it was lost, Russ caught up with the hat exactly one mile downstream from the bridge.
Determine how fast the river was flowing.
Submissions are due by 5:00 PM on Thursday, September 12th. Please send all solutions via email to mathpotw@acu.edu.
Solution to Find the Speed
Correct solutions were submitted by: Wyatt Witemeyer, Connie Yarema, and Bethany Witemeyer
In this solution, we use the formula distance = rate * time repeatedly. Let the speed of the stream be S and the speed that Russ rows in still water be R, both in units of miles per minute. After landing in the stream, the hat travels 1 mile at a rate S, so the time taken is 1/S. During this same amount of time, Russ went upstream 5 minutes, then returned to the bridge, and then continued downstream for a mile. We will formulate an expression for the time in each of Russ’ three phases, add them, and set the sum equal to 1/S.
Going upstream took 5 minutes, and covered a distance D. Coming back to the bridge yields the time D/(R+S). But the distance D can be obtained from D = (R–S)*5. So the time for coming back to the bridge is (R–S)*5/(R+S). Lastly, the time to go the mile from the bridge to the place where Russ caught up with his hat is calculated as 1/(R+S). Finally putting this all together gives the equation
5 + (R–S)*5/(R+S) + 1/(R+S) = 1/S.Algebraic manipulation yields the equation 10RS = R. Since R > 0, we divide to get S = 1/10 miles per minute. This speed is equivalent to 6 miles per hour.
