Question
From an observation point close to the edge of the sea, a ship is 14km away on a bearing of 220 degrees. The distance separating the ships is
Answers
Answered by
GPT-4o mini
To determine the distance separating the observation point from the ship, we can visualize the situation in a coordinate system.
1. Place the observation point at the origin \((0, 0)\) of a coordinate plane.
2. The ship is located 14 km away at a bearing of 220 degrees.
Bearings are measured clockwise from true north. A bearing of 220 degrees corresponds to an angle of \(220^\circ - 180^\circ = 40^\circ\) measured from the south direction.
To convert the bearing to standard angle measurements (counter-clockwise from the positive x-axis), we can find the angle in relation to the Cartesian coordinate system. The angle from the positive x-axis is calculated as:
\[
\theta = 360^\circ - 220^\circ = 140^\circ
\]
The ship's coordinates can be calculated using trigonometric functions:
- The x-coordinate (east-west) is determined by the cosine of the angle:
\[
x = 14 \times \cos(140^\circ)
\]
- The y-coordinate (north-south) is determined by the sine of the angle:
\[
y = 14 \times \sin(140^\circ)
\]
Now, we’ll compute the x and y coordinates:
- \(\cos(140^\circ) = -\cos(40^\circ) \approx -0.7660\)
- \(\sin(140^\circ) = \sin(40^\circ) \approx 0.6428\)
Thus,
\[
x \approx 14 \times (-0.7660) \approx -10.724 \text{ km}
\]
\[
y \approx 14 \times 0.6428 \approx 8.998 \text{ km}
\]
This means the coordinates of the ship are approximately:
\[
(-10.724, 8.998)
\]
The distance from the observation point to the ship is the distance of the point \((-10.724, 8.998)\) from the origin \((0, 0)\):
\[
\text{Distance} = \sqrt{x^2 + y^2} = \sqrt{(-10.724)^2 + (8.998)^2}
\]
Calculating the values:
\[
(-10.724)^2 \approx 115.047 \quad \text{and} \quad (8.998)^2 \approx 80.034
\]
Adding them up:
\[
\sqrt{115.047 + 80.034} = \sqrt{195.081} \approx 13.97 \text{ km}
\]
Thus, the distance separating the observation point from the ship is approximately **14 km** (to a reasonable degree given rounding errors).
1. Place the observation point at the origin \((0, 0)\) of a coordinate plane.
2. The ship is located 14 km away at a bearing of 220 degrees.
Bearings are measured clockwise from true north. A bearing of 220 degrees corresponds to an angle of \(220^\circ - 180^\circ = 40^\circ\) measured from the south direction.
To convert the bearing to standard angle measurements (counter-clockwise from the positive x-axis), we can find the angle in relation to the Cartesian coordinate system. The angle from the positive x-axis is calculated as:
\[
\theta = 360^\circ - 220^\circ = 140^\circ
\]
The ship's coordinates can be calculated using trigonometric functions:
- The x-coordinate (east-west) is determined by the cosine of the angle:
\[
x = 14 \times \cos(140^\circ)
\]
- The y-coordinate (north-south) is determined by the sine of the angle:
\[
y = 14 \times \sin(140^\circ)
\]
Now, we’ll compute the x and y coordinates:
- \(\cos(140^\circ) = -\cos(40^\circ) \approx -0.7660\)
- \(\sin(140^\circ) = \sin(40^\circ) \approx 0.6428\)
Thus,
\[
x \approx 14 \times (-0.7660) \approx -10.724 \text{ km}
\]
\[
y \approx 14 \times 0.6428 \approx 8.998 \text{ km}
\]
This means the coordinates of the ship are approximately:
\[
(-10.724, 8.998)
\]
The distance from the observation point to the ship is the distance of the point \((-10.724, 8.998)\) from the origin \((0, 0)\):
\[
\text{Distance} = \sqrt{x^2 + y^2} = \sqrt{(-10.724)^2 + (8.998)^2}
\]
Calculating the values:
\[
(-10.724)^2 \approx 115.047 \quad \text{and} \quad (8.998)^2 \approx 80.034
\]
Adding them up:
\[
\sqrt{115.047 + 80.034} = \sqrt{195.081} \approx 13.97 \text{ km}
\]
Thus, the distance separating the observation point from the ship is approximately **14 km** (to a reasonable degree given rounding errors).