A ship leaves port with a bearing of S 40 W. After traveling 7 miles, the ship turns 90 degrees on a bearing of N 50 W for 11 miles. At that time, what is the bearing of the ship from port?
using the usual trig orientation, with due East as 0°
S40W = 220°
N50W = 140°
so,starting at (0,0) after the first 7 miles, the ship is at (7cos220°,7sin220°) = (-5.36,-4.50)
Next from there it proceeds another (11cos140°,11sin140°) = (-8.43,7.07)
It is now at (-13.79,2.57)
distance = 14.03 miles
tanx = 2.57/-13.79 = 180-10.55°
bearing from port is W10.55°N
To find the bearing of the ship from the port, we'll break the ship's journey into two parts and calculate each part separately.
1. First, the ship travels with a bearing of S 40 W for 7 miles.
The bearing S 40 W means the ship is heading 40 degrees west of south.
To find the bearing from the port, we need to subtract 180 degrees from the bearing S 40 W:
180 - 40 = 140 degrees
So, after traveling 7 miles, the ship's bearing from the port is N 140 W.
2. Next, after traveling 7 miles, the ship turns 90 degrees on a bearing of N 50 W for 11 miles.
The bearing N 50 W means the ship is heading 50 degrees west of north.
To find the new bearing from the port, we need to subtract 90 degrees from the current bearing N 140 W:
140 - 90 = 50 degrees
So, after traveling 11 miles, the ship's bearing from the port is N 50 W.
Therefore, at that time, the ship's bearing from the port is N 50 W.
To find the bearing of the ship from the port, we need to use the concept of vector addition. Let's break down the ship's journey into two separate legs:
1. The ship travels in the south-west direction (bearing: S 40 W) for 7 miles.
2. After that, the ship makes a 90-degree turn to the left (counterclockwise) and travels in the north-west direction (bearing: N 50 W) for 11 miles.
Now, let's calculate the displacement vectors for each leg of the journey:
1. Displacement vector for the first leg:
- Distance: 7 miles
- Direction: S 40 W
To convert the given bearing to a vector, we can use trigonometry. Since the south direction is opposite to the positive y-axis, and the west direction is negative on the x-axis, we can determine the components of this vector using sine and cosine:
- Horizontal component: 7 * cos(40°) = 7 * (cosine of 40 degrees)
- Vertical component: -7 * sin(40°) = -7 * (sine of 40 degrees)
Therefore, the displacement vector for the first leg is approximately (-4.75, -4.49).
2. Displacement vector for the second leg:
- Distance: 11 miles
- Direction: N 50 W
Using the same approach as before, we can determine the components of this vector:
- Horizontal component: 11 * cos(50°) = 11 * (cosine of 50 degrees)
- Vertical component: 11 * sin(50°) = 11 * (sine of 50 degrees)
Therefore, the displacement vector for the second leg is approximately (-7.02, 8.53).
Now, we can add these two vectors to find the resultant displacement vector:
Resultant displacement vector ≈ (-4.75, -4.49) + (-7.02, 8.53)
To add vectors, simply add their corresponding components:
Resultant displacement vector ≈ (-4.75 - 7.02, -4.49 + 8.53)
Resultant displacement vector ≈ (-11.77, 4.04)
To find the resultant bearing of the ship from the port, we can use trigonometry again:
Resultant bearing ≈ arctan(4.04 / -11.77)
Using a scientific calculator or an online tool, we can find the inverse tangent of the ratio to calculate the angle:
Resultant bearing ≈ -19.32°
Therefore, at that time, the bearing of the ship from the port is approximately N 19° 19' W.