Final answer:
The rate of the river's current is found to be 14 mph by setting the times to travel a certain distance downstream and upstream equal and solving for the current speed.
Step-by-step explanation:
The student's question requires us to find the rate of the river's current given that a motorboat travels at different speeds upstream and downstream. To solve the problem, we'll use the concept of relative velocity in still water versus moving water.
Step-by-Step Solution
Let the speed of the current be x mph. The boat's speed downstream, which is the speed in still water plus the current, is (35 + x) mph. Upstream, the boat's speed is the speed in still water minus the current, so it's (35 - x) mph.
Since it takes the same amount of time to travel 7 miles downstream as it does to travel 3 miles upstream, we can use the equation time = distance / speed. Therefore, the time to travel downstream is 7 / (35 + x) hours, and the time to travel upstream is 3 / (35 - x) hours. Setting these two times equal, as per the problem, we get:
7 / (35 + x) = 3 / (35 - x)
Cross multiplying and solving for x, we get:
7(35 - x) = 3(35 + x)
245 - 7x = 105 + 3x
245 - 105 = 7x + 3x
140 = 10x
x = 14
Therefore, the rate of the river's current is 14 mph.