# a ship starts from port at a bearing of South 32 degrees East. It travels 8 miles in this direction before turning to a bearing of North 14 degrees East. It travels 15 miles in this direction, before finally turning to a bearing of South 72 degrees East and travels another 4 miles in this direction. How far is the ship from the port?

## Given: AB = 8mi[148o]CW, BC = 15[14o], CD = 4[108o].

X = 8*sin148+15*sin14+4*sin108 = 11.7 miles.
Y = 8*cos148+15*cos14+4*cos108 = 6.5i miles.
D = 11.7 + 6.5i = 13.4mi[61] CW.

## D = 11.7 +6.5i = 13.4mi[61o] = AD.

DA = 13.4[61+180] = 13.4mi[241o] CW = bearing from final point to port.

## To find the distance of the ship from the port, we can break down the ship's journey into different segments and use trigonometry to calculate the distances.

1. Ship's first segment: It travels 8 miles on a bearing of South 32 degrees East.
To calculate the horizontal distance covered, we need to find the component of the distance traveled in the eastward direction.
Eastward distance = 8 * cos(32 degrees)
Eastward distance ≈ 6.78 miles (rounded to two decimal places)

To calculate the vertical distance covered, we need to find the component of the distance traveled in the southward direction.
Southward distance = 8 * sin(32 degrees)
Southward distance ≈ 4.23 miles (rounded to two decimal places)

2. Ship's second segment: It travels 15 miles on a bearing of North 14 degrees East.
We need to find the eastward and northward distances.
Eastward distance = 15 * cos(14 degrees)
Eastward distance ≈ 14.69 miles (rounded to two decimal places)

Northward distance = 15 * sin(14 degrees)
Northward distance ≈ 3.96 miles (rounded to two decimal places)

3. Ship's third segment: It travels 4 miles on a bearing of South 72 degrees East.
We need to find the eastward and southward distances.
Eastward distance = 4 * cos(72 degrees)
Eastward distance ≈ 1.30 miles (rounded to two decimal places)

Southward distance = 4 * sin(72 degrees)
Southward distance ≈ 3.86 miles (rounded to two decimal places)

Now, let's add up all the eastward and westward distances as well as the northward and southward distances:

Eastward distance = 6.78 + 14.69 + 1.30 ≈ 22.77 miles (rounded to two decimal places)
Northward distance = 3.96
Southward distance = 4.23 + 3.86 ≈ 8.09 miles (rounded to two decimal places)

To find the overall distance from the port, we can use the Pythagorean theorem:
Distance = √(Eastward distance^2 + (Northward distance - Southward distance)^2)
Distance = √(22.77^2 + (3.96 - 8.09)^2)
Distance ≈ √(519.43 + (-16.32)^2)
Distance ≈ √(519.43 + 266.13)
Distance ≈ √(785.56)
Distance ≈ 28 miles (rounded to two decimal places)

Therefore, the ship is approximately 28 miles away from the port.