To find the bearing of point X from point Y, let's break down the problem step-by-step.
1. Understand the positions involved:
- The student starts from a point that is 400 meters north of point X.
- From this starting position, the student walks eastwards to reach point Y.
- Point Y is 800 meters from point X.
2. Determine the respective coordinates:
- Let's consider the origin (0, 0) to be point X.
- The starting position of the student is 400 meters north of X, so it has coordinates (0, 400).
- The distance between point X and the point Y is given as 800 meters.
3. Calculate relative distances:
- After traveling eastwards to point Y from the initial point, the student covers the eastward distance.
- Since the total distance from point X to Y is 800 meters, we need to find a right triangle where one leg is the initial northward distance (400 meters) and the hypotenuse (the straight line from X to Y) is 800 meters.
4. Using Pythagoras’ theorem:
- To find the eastward distance (let's call it [tex]\(d\)[/tex]), we use the Pythagoras theorem:
[tex]\[
d^2 + 400^2 = 800^2
\][/tex]
- Solving for [tex]\(d\)[/tex]:
[tex]\[
d = \sqrt{800^2 - 400^2}
\][/tex]
[tex]\[
d = \sqrt{640000 - 160000}
\][/tex]
[tex]\[
d = \sqrt{480000}
\][/tex]
[tex]\[
d = 800\sqrt{3}/2
\][/tex]
[tex]\[
d ≈ 692.82 \text{ meters}
\][/tex]
5. Bearings:
- The bearing from a point is typically measured clockwise from the north direction.
- To find the bearing of X from Y, we need to calculate the angle from the north line down to the line connecting Y to X.
6. Calculate the tangent of the angle:
- The eastward distance from the student’s starting position to point X is 800 meters.
- The northward distance (between point Y and X) is 400 meters.
- Let θ be the angle we need to find, which is given by:
[tex]\[
\theta = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \arctan\left(\frac{800}{400}\right)
\][/tex]
- Simplifying:
[tex]\[
\theta = \arctan(2)
\][/tex]
7. Convert radians to degrees:
- The tangent inverse of 2 is approximately 1.1071487177940904 radians.
- To convert this to degrees:
[tex]\[
\text{Degrees} = \theta \cdot \left(\frac{180}{\pi}\right)
\][/tex]
which gives approximately 63.43494882292201 degrees.
Therefore, the bearing of X from Y is approximately [tex]\(63.43\)[/tex] degrees clockwise from the north.
