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?
I recommend doing this graphically.
I haven tried doing it graphically using the law of sines but am still unable to come up with a consistent answer. It is also asking what is the bearing of the ship to the port from the final point?
graphically is not the law of sines...
ok thanks for the help?
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.