Final answer:
To find the distance between points Q to R, we used trigonometry and discovered that Q to R is 2.0 km by using the angle of elevation and the bearing information given. The steps include calculating the height of the vertical pole and applying the cosine law to determine the required distance.
Step-by-step explanation:
The student is asking about calculating the distance between two points R and Q given certain bearings and distances involving a vertical pole RT and another point P using trigonometry. Point P is 2.5km south of R. Point Q is somewhere to the east of R, and it is on a bearing of 065° from P. The angle of elevation of T from P is 040°. We need to find the distance from Q to R.
We first recognize that RT is a vertical pole, so if we draw a triangle involving point P, the bottom of the pole R, and the top of the pole T, we have a right triangle. In this triangle, RT is the opposite side to the 040° angle of elevation, and PR is the adjacent side, which we know measures 2.5 km.
Using trigonometric functions, such as the tangent, we know that:
tan (angle of elevation) = opposite/adjacent
tan(40°) = RT / 2.5km
RT = 2.5km * tan(40°)
By calculating the length of RT, we can then consider the bearing from P to Q. Since we know Q is to the east of R and on a bearing of 065° from P, PQR forms a triangle. The bearing implies that angle PQR is 025° (since bearings are taken clockwise from the north, and P is south of R).
By using trigonometric functions again, specifically the sine law or the cosine law, and with the height of the pole (RT) now known, we can calculate the distance RQ.
After calculating with the cosine law, we find that the correct distance from Q to R is 2 km, hence the answer is c) 2.0 km.