A boat travels 80 miles each way downstream and

back. The total trip took 15 hours. If the boat travels

at 12 miles per hour in still water, what is the speed

of the current in miles per hour? Use rational

equations to solve this problem.

The speed of the current is 4 mph.

Step-by-step explanation:

The speed of the boat in still water is 12 mph, but when it's going against the stream it is "12 - x" mph and when it's going with the stream it is "12 + x" mph. Since the total trip took 15 h, then the sum of the times from each leg of the trip must be equal to that value. Using the average speed formula, we can manipulate it to give us the time of each leg as shown below:

speed = distance / time

time*speed = distance

time = distance / speed

For the downstream:

time 1 = 80 / (12 + x)

For the upstream:

time 2 = 80 / (12 - x)

The sum of these two times must be equal to 15 h, therefore:

15 = [80 / (12 + x) ] + [80 / (12 - x) ]

15 = [80 * (12 - x) + 80 * (12 + x) ]/[ (12+x) * (12-x) ]

15 = {80*[ (12 - x) + (12 + x) ]}/[12² - x²]

15 = {80*[12 + 12 - x + x]} / (144-x²)

15 = (80*24) / (144 - x²)

15 = 1920 / (144 - x²)

15 * (144 - x²) = 1920

2160 - 15x² = 1920

-15x² = 1920 - 2160

-15x² = - 240

x² = - 240 / - 15 = 16

x = 4 or x = - 4

Since the speed can't be negative in this context, the speed of the current is 4 mph.