P and Q are two observation posts on the same horizontal ground asthe foot R of a vertical pole RT. P is 2.5km south of R. Q is east of R and is on a bearing of 065° from P. The angle of elevation of T from P is 40°.calculate, correct to three significant figures The distance PQ
First, we can draw a diagram to visualize the situation:
```
T
/|
/ |
/ |
/ | 40°
/ |
/ |
/ |
R |
| /
| /
| /
| /
| /
| /
|/
P
```
We can see from the diagram that we have a right triangle PRQ, where ∠P = 90°, PR = 2.5 km, and ∠Q = 65°.
To find the distance PQ, we need to find the length of QR. We can use trigonometry (specifically the sine function) to do this.
In triangle PRQ, we have:
sin Q = QR/PR
Rearranging the equation:
QR = PR * sin Q
Substituting the given values:
QR = 2.5 km * sin 65°
Using a calculator, we find:
QR ≈ 2.5 km * 0.90631
QR ≈ 2.26578 km
Therefore, the distance PQ is approximately 2.266 km.