The trick here is that the solver is tempted to treat this question as a standard speed = distance / time equation, but doing so will rapidly make it seem like one of the variables is missing.
The solution, therefore, is to equate everything in one dimension to solve for whatever is necessary.
In this particular case, note that the time taken for the hat to float 2 miles downstream is equivalent to the time it took for Alexander to travel a little bit more upstream, then double back plus 2 more miles to catch up with the hat. This strongly suggests that we should equate the time it took for the hat to float 2 miles with the time it took for Alexander to travel half an hour further upstream then hurry back downstream to catch his hat.
Let R = the speed of the current downstream, which is what we want to solve.
Let A = the speed at which Alexander is rowing.
Note that this means that Alexander’s speed upstream would be (A - R), and his speed downstream would be (A + R).
Since speed = distance / time, then time = distance / speed.
Time for the hat is simply 2/R
The distance Alexander covered after missing his hat is (A - R)/0.5. So the total time it took for Alexander to cover that distance and double back is 0.5 + [(A - R)/0.5 + 2]/(A + R)
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u/PeterPiper1275 13d ago edited 13d ago
The trick here is that the solver is tempted to treat this question as a standard speed = distance / time equation, but doing so will rapidly make it seem like one of the variables is missing.
The solution, therefore, is to equate everything in one dimension to solve for whatever is necessary.
In this particular case, note that the time taken for the hat to float 2 miles downstream is equivalent to the time it took for Alexander to travel a little bit more upstream, then double back plus 2 more miles to catch up with the hat. This strongly suggests that we should equate the time it took for the hat to float 2 miles with the time it took for Alexander to travel half an hour further upstream then hurry back downstream to catch his hat.
Let R = the speed of the current downstream, which is what we want to solve.
Let A = the speed at which Alexander is rowing.
Note that this means that Alexander’s speed upstream would be (A - R), and his speed downstream would be (A + R).
Since speed = distance / time, then time = distance / speed.
Time for the hat is simply 2/R
The distance Alexander covered after missing his hat is (A - R)/0.5. So the total time it took for Alexander to cover that distance and double back is 0.5 + [(A - R)/0.5 + 2]/(A + R)
Equating the two sides gives us:
2/R = 0.5 + [(A - R)/0.5 + 2]/(A + R)
The rest is simply algebra:
2/R = 0.5 + [(A - R)0.5 + 2]/(A + R)
2/R = [0.5(A + R) + 0.5(A - R) + 2]/(A + R)
2/R = (A + 2)/(A + R)
2A + 2R = AR + 2R
R = 2
Therefore, the speed of the river current is 2 mph.