Final answer:
The rate of the current, when a boat that travels 36 mph in still water can go 100 miles with the current and 80 miles against the current in the same time, is 4 mph.
Step-by-step explanation:
To find the rate of the current, r, for a boat that can travel 36 mph in still water, we set up two separate equations based on the boat's speed with and against the current. The boat's speed with the current is (36 + r) mph, and against the current is (36 - r) mph. Given that the boat travels 100 miles with the current and 80 miles against the current in the same amount of time, we can establish the following equations based on distance equals speed multiplied by time:
100 = (36 + r)t
80 = (36 - r)t
By dividing the two equations, we can eliminate the variable t (time):
100 / (36 + r) = 80 / (36 - r)
By cross-multiplying and simplifying, we get:
100(36 - r) = 80(36 + r)
3600 - 100r = 2880 + 80r
720 = 180r
r = 720 / 180
r = 4
Therefore, the rate of the current is 4 mph.