Q: A ship leaves the port of LA and travels 120 miles on a bearing of S47ºW and then 110 miles on a bearing of N80ºW. Find the total distance traveled South and West and also its bearing from the port of LA.
I have found the distance traveled South and West (approx. 62.74mi and 196.09mi, respectively), but I don't know how to find the baring in this type of problem.
I assume you can find the normal trig angle associated with the point.
For an angle θ, measured clockwise, the bearing is 90+θ.
Just plot the point and note how many degrees clockwise from vertical it is. That is the compass heading.
To find the bearing of the ship from the port of LA, you can use trigonometry and vector addition.
First, let's break down the ship's journey into its north-south and east-west components:
Distance traveled south = 120 miles * sin(47º) ≈ 81.35 miles
Distance traveled west = 120 miles * cos(47º) ≈ 87.27 miles
Distance traveled north = 110 miles * sin(80º) ≈ 107.06 miles
Distance traveled west = 110 miles * cos(80º) ≈ 24.60 miles
Now, let's add these components to find the total north-south and east-west distances:
Total distance traveled south = 81.35 miles - 107.06 miles ≈ -25.71 miles (since North is positive and South is negative)
Total distance traveled west = 87.27 miles + 24.60 miles ≈ 111.87 miles (since West is positive)
The total distance traveled south and west is approximately 25.71 miles south and 111.87 miles west.
To find the bearing of the ship from the port of LA, we can use the tangent function:
Bearing = tan^(-1)(Total distance traveled west / Total distance traveled south)
Bearing = tan^(-1)(111.87 miles / -25.71 miles) ≈ -77.8º
Since the bearing is negative, we add 180º to find the bearing in the direction of the port of LA:
Bearing from the port of LA = -77.8º + 180º ≈ 102.2º
Therefore, the ship's bearing from the port of LA is approximately N102.2ºW.