Final answer:
The speed of the current is found by calculating the effective speed of the tugboat upstream and downstream using its speed in still water and the distances traveled. By setting up an equation based on the total travel time and solving for the current's speed, we find that it is approximately 11.8 miles per hour.
Step-by-step explanation:
To find the speed of the current, we can use the information given about the tugboat's speed in still water and the time it takes to travel upstream and downstream. The tugboat's speed in still water is 13 miles per hour. Let's denote the speed of the current as c.
When the tugboat travels upstream against the current, its effective speed is reduced by the current's speed, which is (13 - c) miles per hour. Similarly, when the tugboat travels downstream with the current, its effective speed is increased by the current's speed, which is (13 + c) miles per hour. We know that the distance traveled in both cases is 39 miles and the total time for both trips is 6.18 hours.
- Calculate the time to travel upstream: time_upstream = distance_upstream / speed_upstream = 39 / (13 - c).
- Calculate the time to travel downstream: time_downstream = distance_downstream / speed_downstream = 39 / (13 + c).
- The total time is the sum of time_upstream and time_downstream: 6.18 hours = 39 / (13 - c) + 39 / (13 + c).
- Solve the above equation for c to find the current's speed.
Using algebra, we rearrange the equation to solve for c:
6.18 = 39 / (13 - c) + 39 / (13 + c)
6.18(13 - c)(13 + c) = 39(13 + c) + 39(13 - c)
80.34 - 6.18c^2 = 507 + 507
-6.18c^2 = 933.66 - 80.34
-6.18c^2 = 853.32
c^2 = -853.32 / -6.18
c = √(853.32 / 6.18)
c ≈ 11.8
Thus, the speed of the current is approximately 11.8 miles per hour.