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

To find the time it takes for the boat to return to port, we need to calculate the distances covered during each segment of the journey and then add them up.

First, let's calculate the distance covered during the first segment. The boat travels at a speed of 5 miles per hour for 2 hours. So, the distance covered during this segment is:

Distance_1 = Speed_1 * Time_1
Distance_1 = 5 mph * 2 hours
Distance_1 = 10 miles

Next, let's calculate the distance covered during the second segment. The boat travels at a speed of 12 miles per hour for 2 hours. So, the distance covered during this segment is:

Distance_2 = Speed_2 * Time_2
Distance_2 = 12 mph * 2 hours
Distance_2 = 24 miles

Finally, let's calculate the distance covered during the third segment. The boat travels at a speed of 10 miles per hour for 5 hours. So, the distance covered during this segment is:

Distance_3 = Speed_3 * Time_3
Distance_3 = 10 mph * 5 hours
Distance_3 = 50 miles

Now, let's calculate the total distance covered during the journey by adding up the distances covered in each segment:

Total Distance = Distance_1 + Distance_2 + Distance_3
Total Distance = 10 miles + 24 miles + 50 miles
Total Distance = 84 miles

After traveling 84 miles, the boat changes its bearing to 180∘ and heads directly back to the port at a speed of 3 miles per hour. To find the time it takes to return, we need to divide the total distance by the speed of the boat during the return journey:

Time to Return = Total Distance / Speed_Return
Time to Return = 84 miles / 3 mph
Time to Return ≈ 28 hours

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

As for the boat's bearing as it returns, since it changes to a bearing of 180∘ for the return journey, it is heading directly opposite to the initial bearing. So, the boat's bearing as it returns to port is 350∘ + 180∘ = 530∘.