# 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.

## d1 = 5mi/h[350o] * 2h = 10mi[350o]

d2 = 12mi/h[40o] * 2h = 24mi[40o]

d3 = 10mi/h[180] * 5h = 50mi[180]

X = 10*Cos350 + 24*Cos40 + 50*Cos180 =

-21.8 Miles.

Y = 10*sin350 + 24*sin40 + 50*sin180 =

13.7 Miles.

d = d1+d2+d3 = X + Yi=-21.8 + 13.7i,Q2.

Tan Ar = Y/X = 13.7/-21.8 = -0.62800

Ar = -32.13o = Reference angle.

A = -32.13 + 180 = 147.9o, CCW.

d = X/cosA = -21.8/Cos147.9 = 25.7 Miles

[147.9o]

T = d/V = 25.7mi[147.9o]/3 = 8.57 Hours.

Direction = 147.9 Degrees.

## To find the time it takes for the boat to return to port, we need to calculate the total distance traveled and then divide it by the speed at which the boat is returning to port.

Let's break down the boat's journey and calculate the distances traveled:

1. From the starting point, the boat travels at a speed of 5 miles per hour for 2 hours in the direction of a bearing of 350∘.

- Distance traveled = Speed * Time = 5 miles/hour * 2 hours = 10 miles.

2. The boat then changes its bearing to 40∘ and continues at a speed of 12 miles per hour for 2 hours.

- Distance traveled = Speed * Time = 12 miles/hour * 2 hours = 24 miles.

3. Next, the boat changes its bearing to 180∘ (directly opposite the starting point) and continues at a speed of 10 miles per hour for 5 hours.

- Distance traveled = Speed * Time = 10 miles/hour * 5 hours = 50 miles.

So far, the boat has traveled a total distance of 10 miles (first leg) + 24 miles (second leg) + 50 miles (third leg) = 84 miles.

Now, the boat heads directly back to the port at a speed of 3 miles per hour. We can calculate how long it takes for the boat to return:

- Time taken to return = Distance / Speed = 84 miles / 3 miles/hour = 28 hours.

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

To find the boat's bearing as it returns to port, we can determine the angle between the starting point and the port, which would be the bearing we're looking for. We know that the starting bearing was 350∘, and the boat has made a 180∘ change in direction to head back to port.

- Bearing = Starting bearing + 180∘ = 350∘ + 180∘ = 530∘.

Therefore, the boat's bearing as it returns to port is 530∘.