# A fishing boat leaves port at 5 miles per hour at a bearing of 350∘ for 2 hours, then turns to a bearing of 40∘ at 12 miles per hour for 2 hours, and finally changes to a bearing of 180∘ at 10 miles per hour for 5 hours. At this point, the boat heads directly back to port at a speed of 3 miles per hour. Find the time it takes the boat to return to port as well as the boat's bearing as it does.

## First, it's "heading" or "course," **not** "bearing."

Just figure the x- and y-displacements for each leg of the trip.

Add them up for a final location.

Then figure the distance and (yes) __bearing__ of the port from there.

What do you get? I'll check your answer, or 'splain further if there's some step you can't figure out.

## To find the time it takes for the boat to return to port, we need to calculate the total distance the boat travels away from the port and then divide it by the speed at which it returns.

Let's break down the boat's movements step by step.

First leg: The boat travels at a speed of 5 miles per hour for 2 hours at a bearing of 350∘.

Distance = Speed * Time = 5 mph * 2 hours = 10 miles.

Second leg: The boat turns to a bearing of 40∘ and travels at a speed of 12 miles per hour for 2 hours.

Distance = Speed * Time = 12 mph * 2 hours = 24 miles.

Third leg: The boat changes its bearing to 180∘ and travels at a speed of 10 miles per hour for 5 hours.

Distance = Speed * Time = 10 mph * 5 hours = 50 miles.

Now, let's find the total distance the boat travels away from the port:

Total distance = Distance of first leg + Distance of second leg + Distance of third leg

Total distance = 10 miles + 24 miles + 50 miles = 84 miles.

Finally, to find the time it takes for the boat to return to the port:

Time = Distance / Speed = 84 miles / 3 mph = 28 hours.

Therefore, it takes the boat 28 hours to return to the port.

To determine the boat's bearing as it returns to port, we need to consider the last leg of the journey, which is directly back to port. The boat's bearing is given as 180∘, indicating it is heading in the opposite direction from where it started. Thus, the boat's bearing as it returns to port is 180∘.