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
To find the distance PQ, we can use trigonometry.
First, we need to find the distance PR. Since P is 2.5 km south of R, the distance PR is also 2.5 km.
Next, we can use the angle of elevation of T from P to find the height of the pole RT. We can use the tangent function:
tan(40°) = height/PR
tan(40°) = height/2.5
height = tan(40°) * 2.5
Using a calculator, we find that the height is approximately 2.108 km.
Now, we can use the height and the bearing of Q from P to find the distance PT. We can use the sine and cosine functions:
sin(65°) = height/PT
cos(65°) = PQ/PT
Substituting the known values, we have:
sin(65°) = 2.108/PT
cos(65°) = PQ/PT
Solving the first equation for PT, we find:
PT = 2.108/sin(65°)
Solving the second equation for PQ, we find:
PQ = PT * cos(65°)
Using a calculator, we can evaluate these equations to find:
PT ≈ 2.389 km
PQ ≈ 0.977 km
Therefore, the distance PQ is approximately 0.977 km, correct to three significant figures.