Originally posted by Jirakon
Al, Ben, Carl, and Dan need to get to the special place. They are currently 20 miles from the special place, and they can each walk at a constant 4 miles per hour. They figure they can get there in 5 hours, but it would behoove them to get there as soon as possible.
Just then, they meet a friendly motorcyclist named Ed who can travel at 56 miles per hour, ...[text shortened]... t take for the five of them reach the special place, and what was the initial distance x?
The motocycle will make 7 trips total, call them M1, M2, M3, M4, M5, M6, M7
Since the motorcycle and the walkers are moving at constant speeds, M1 = M3 = M5 = M7, and M2 = M4 = M6
so the motorcycle will be travels two different distances, M1 and M2
While the motorcycle is traveling M1, the walkers are walking W1.
W2 corresponds to the distance the walkers travel while the the motorcycle goes M2.
Now that we have our variables, we can lay out equations.
since 56/4 = 14,
M1 = 14*W1
If the motorcycle travels out and back 14 miles and meets the walkers a mile from where they started, the motorcycle can go
x + y = 14, and
x - y = 1
therefore, we know x = 7.5, so
M1 = 7.5 (W1 + W2)
solving for W2, we get W2 = 13/15 * W1
finally, we know that M1 + 3 * (W1 + W2) = 20
14*W1 + 3 * W1 + 39/15 * W1 = 20
therefore W1 ~= 1.021 miles, and
M1 = x = 14.286 miles
*edit* to be more precise, W1 = 50/49 miles, and M1 = 700/49 miles
the time taken is (15 min/mile) * (20 - 700/49 miles) = 1:25:42.86