Final answer:
The rate of the boat in still water is found by setting up the equation \( \frac{25}{x + 5} = \frac{15}{x - 5} \), where x represents the boat speed in still water and 5 mph is the current speed. After cross-multiplying and solving this equation, we find that the boat's rate in still water is 20 mph.
Step-by-step explanation:
To answer the question "What’s the boat's rate in still water?", we need to use the concept of relative velocity. The current of the river affects the overall speed of the boat when going downstream and upstream. The rate of the current is given as 5 mph.
When the boat is going downstream, its speed is the sum of its still water speed (boat speed in still water) and the current speed. Conversely, when traveling upstream, the speed of the current subtracts from the speed of the boat.
Let the boat's rate in still water be x mph. When going downstream, its speed becomes x + 5 mph because of the downstream current, and when going upstream, it becomes x - 5 mph.
The time it takes to travel a certain distance is the distance divided by the speed. Since the boat travels 25 miles downstream and 15 miles upstream in the same amount of time, we have the following equation: \( \frac{25}{x + 5} = \frac{15}{x - 5} \).
To find x, we cross-multiply and solve the resulting equation:
- 25(x - 5) = 15(x + 5)
- 25x - 125 = 15x + 75
- 25x - 15x = 75 + 125
- 10x = 200
- x = 20 mph
The boat's rate in still water is 20 mph.