Let's assume the rate of the motorboat in still water is represented by "b" (in miles per hour) and the rate of the current is represented by "c" (in miles per hour).
When the motorboat is traveling upstream, i.e., against the current, its effective speed is reduced. The speed of the current subtracts from the speed of the boat. So, the effective speed of the boat traveling upstream is (b - c) miles per hour.
Similarly, when the motorboat is traveling downstream, i.e., with the current, the speed of the current adds to the speed of the boat. So, the effective speed of the boat traveling downstream is (b + c) miles per hour.
Given the following information:
- The boat travels 132 miles in 3 hours upstream at a speed of (b - c) miles per hour.
- The boat travels 162 miles in 3 hours downstream at a speed of (b + c) miles per hour.
We can set up the following equations based on the distance formula (distance = speed × time):
Upstream: 132 = (b - c) × 3 Downstream: 162 = (b + c) × 3
Let's solve these equations to find the values of "b" and "c."
From the first equation: 3(b - c) = 132 b - c = 44 (dividing both sides by 3)
From the second equation: 3(b + c) = 162 b + c = 54 (dividing both sides by 3)
Now, we have a system of equations:
b - c = 44 b + c = 54
Adding these equations together eliminates "c": 2b = 98 b = 49
Substituting the value of "b" back into one of the equations: 49 + c = 54 c = 5
Therefore, the rate of the motorboat in still water is 49 miles per hour, and the rate of the current is 5 miles per hour.