A surveyor standing at a point X sites a pole Y due east of him and a tower X of a building on a bearing of 046 degrees, after walking to a point W, a distance of 180m on the south east direction, he observes a bearing of Z and Y to be 337 degrees and 050 degrees respectively . Calculate to the nearest meter line XY
I think your diagram is in error, since W is SE of X.
In triangle XYW, we have
X=45°, Y=40°, W=72°
So now we have
XY/sin72° = 180/sin40°
XY = 266.32
You are correct, I apologize for the error in my diagram and calculation. Thank you for catching that and providing the correct solution.
To find the length of line XY, we can use trigonometry and the information given.
1. From the information given, we know that the bearing from point X to the tower X is 046 degrees.
2. We also know that after walking to point W, the surveyor observes a bearing of Z and Y to be 337 degrees and 050 degrees respectively.
3. To find the bearing between points X and W, we can subtract the observed bearing of Z from 360 degrees since bearings are measured clockwise from the north.
Bearing(XW) = 360 degrees - Z
= 360 degrees - 337 degrees
= 23 degrees
4. Now, we can consider a triangle with sides XW, XY, and YW.
5. We can use the sine rule to calculate XY:
sin(23 degrees) = XY / 180m (using the distance from X to W)
XY = sin(23 degrees) * 180m
6. Use a calculator to find the value of sin(23 degrees) and multiply it by 180m to find the length of XY.
XY ≈ 68.76m
So, line XY is approximately 68.76 meters long.
First, we can draw a diagram to represent the situation:
```
W
/|
/ |
/ | 180m
/ |
/ Z |
/_____|
X Y Building
```
We are given that XY is due east, so we can draw a line due east from X:
```
W
/|
/ |
/ | 180m
/ |
/ Z |
/_____|
X----->Y Building
```
We can label the angle at X as 46 degrees (since the bearing to the building is 046 degrees). We can also label the angle at Y as 10 degrees (since the bearing from W to Y is 050, which is 40 degrees clockwise from due north). Finally, we can label the angle at Z as 23 degrees (since the bearing from W to Z is 337, which is 23 degrees clockwise from due north).
Now, we can use the Law of Sines to solve for XY:
```
sin(46) / XY = sin(10) / 180
sin(23) / XY = sin(147) / 180
```
Solving for XY in either equation gives:
```
XY = sin(46) * 180 / sin(10) ≈ 1026.6m
XY = sin(23) * 180 / sin(147) ≈ 1026.6m
```
So, to the nearest meter, line XY is approximately 1027m long.