einstein10
Points A and B are 120 km apart. A motorcyclist starts from A to B along straight road AB with speed 30 kmph. At the same time a cyclist starts from B along a road perpendicular to road AB, with a speed of 10 kmph. After how many hours will the distance between them be the least?
A. 3 hours
B. 3.4 hours
C. 3.5 hours
D. 3.6 hours
E. None
Let M = the motorcyclist and C = the cyclist.
After 3.4 hours:
M has traveled 102 km of the 120 km between A and B, placing him 18 km from B.
C has traveled 34 km from B.
Every 1/10 of an hour, M's distance from B decreases by 3 km, while the C's distance from B increases by 1 km, as follows:
.....0 hr....3.4 hrs....3.5 hrs....3.6 hrs....3.7 hrs
M: 120--->18------>
15-------->
12------>
9C: 000--->34------->
35------->
36------>
37Since M and C travel along perpendicular roads, the distance between them constitutes the hypotenuse of a right triangle, as follows:
\((distance)^2 =\) \(M^2 + C^2\)
The colored combinations above imply the following:
\(M^2 + C^2\) =
\(15^2 + 35^2 = 1450\)\(M^2 + C^2\) =
\(12^2 + 36^2 = 1440\)\(M^2 + C^2\) =
\(9^2 + 37^2 = 1450\)The results in red are equal and exceed the result in green.
Implication:
The least distance will be yielded by the values in green, which occur at the 3.6-hour mark.
The following problem is similar but involves much friendlier values:
https://gmatclub.com/forum/andrew-start ... 21827.html