2011 AMC 12A Problem 12

Question

A power boat and a raft both left dock

A

on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock

B

downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river

9

hours after leaving dock

A.

How many hours did it take the power boat to go from

A

to

B?

3

3.5

4

4.5

5

Answer: D

Concepts:relative speed distance rate and time symmetry

Difficulty rating: 1580

Small hint:

Work in the frame of the water, in which the raft stays put

Big hint:

Relative to the water the boat travels at the same speed going out and coming back, so it takes equal times each way

Solution:

Measure everything relative to the water. In that frame the raft is stationary at the point where the boat started, and the boat moves at its constant speed

v

relative to the water, both downstream and upstream.

The boat leaves the raft, travels away for some time, then returns to it at the same relative speed, so it spends equal times going and returning. Hence the outbound leg to

B

takes half of

9,

which is

4.5

hours.

Thus, the correct answer is D.