Let's break down the problem step-by-step and find the required values based on the given conditions:
### Given Data:
- Point [tex]\(A\)[/tex] is at a bearing of [tex]\(120^\circ\)[/tex] from point [tex]\(P\)[/tex] and is [tex]\(100 \, \text{m}\)[/tex] away from [tex]\(P\)[/tex].
- Point [tex]\(R\)[/tex] is at a bearing of [tex]\(45^\circ\)[/tex] from point [tex]\(A\)[/tex] and is [tex]\(150 \, \text{m}\)[/tex] away from [tex]\(A\)[/tex].
### Step-by-Step Solution:
1. Convert bearings to Cartesian coordinates:
- Bearings are typically measured clockwise from the north. We will convert these bearings to radians for coordinate calculations.
2. Calculate the coordinates of point [tex]\(A\)[/tex] relative to [tex]\(P\)[/tex]:
- [tex]\(x\)[/tex]-coordinate of [tex]\(A\)[/tex], [tex]\(A_x = 100 \cdot \cos(120^\circ)\)[/tex]
- [tex]\(y\)[/tex]-coordinate of [tex]\(A\)[/tex], [tex]\(A_y = 100 \cdot \sin(120^\circ)\)[/tex]
Given the result:
- [tex]\( A_x \approx -50.00 \)[/tex]
- [tex]\( A_y \approx 86.60 \)[/tex]
3. Calculate the coordinates of point [tex]\(R\)[/tex] relative to [tex]\(A\)[/tex]:
- [tex]\(x\)[/tex]-coordinate of [tex]\(R\)[/tex] from [tex]\(A\)[/tex], [tex]\(R_x' = 150 \cdot \cos(45^\circ)\)[/tex]
- [tex]\(y\)[/tex]-coordinate of [tex]\(R\)[/tex] from [tex]\(A\)[/tex], [tex]\(R_y' = 150 \cdot \sin(45^\circ)\)[/tex]
Given the result:
- [tex]\( R_x' \approx 106.07 \)[/tex]
- [tex]\( R_y' \approx 106.07 \)[/tex]
4. Calculate the coordinates of point [tex]\(R\)[/tex] relative to [tex]\(P\)[/tex]:
- [tex]\(R_x = A_x + R_x'\)[/tex]
- [tex]\(R_y = A_y + R_y'\)[/tex]
Given the result:
- [tex]\( R_x \approx 56.07 \)[/tex]
- [tex]\( R_y \approx 192.67 \)[/tex]
5. Calculate the distance between point [tex]\(P\)[/tex] and point [tex]\(R\)[/tex]:
- Distance [tex]\(PR = \sqrt{(R_x)^2 + (R_y)^2}\)[/tex]
Given the result:
- [tex]\(PR \approx 200.66 \)[/tex]
6. Calculate the bearing of point [tex]\(P\)[/tex] from point [tex]\(R\)[/tex]:
- Bearing [tex]\(RP = \text{atan2}(-R_y, -R_x)\)[/tex] (convert to degrees)
Given the result:
- [tex]\(RP \approx 253.78^\circ\)[/tex]
### Summary:
- Coordinates of point [tex]\(A\)[/tex] relative to [tex]\(P\)[/tex]:
- [tex]\(A_x \approx -50.00\)[/tex]
- [tex]\(A_y \approx 86.60\)[/tex]
- Coordinates of point [tex]\(R\)[/tex] relative to [tex]\(A\)[/tex]:
- [tex]\(R_x' \approx 106.07\)[/tex]
- [tex]\(R_y' \approx 106.07\)[/tex]
- Coordinates of point [tex]\(R\)[/tex] relative to [tex]\(P\)[/tex]:
- [tex]\(R_x \approx 56.07\)[/tex]
- [tex]\(R_y \approx 192.67\)[/tex]
- Distance between point [tex]\(P\)[/tex] and point [tex]\(R\)[/tex]:
- [tex]\( \approx 200.66 \, \text{m} \)[/tex]
- Bearing of point [tex]\(P\)[/tex] from point [tex]\(R\)[/tex]:
- [tex]\( \approx 253.78^\circ\)[/tex]
Thus, the calculations reveal the necessary details about the distances and bearings between the points P, A, and R.
