Trigonometry Question - HELP

Question

A river running due east has straight parallel banks. A vertical post stands with its base, P, on the north side of the river. On the south bank are two surveyors, A who is to the east and B who is to the west of the post. A & B are at a distance 2/7a apart and the angle APB = 150 degrees. The angles of elevation from A and B of the top Q, of the post are 45 degrees and 30 degrees. Find in terms of a, the width of the river and the height of the post.

Answer 1

This is actually two related problems. We should first draw representations of each scenario, so that we are in a position to visualize each situation & solve each problem. We also must make a few assumptions. The problem does not state that the post is exactly on the edge of the river, but we should assume this. The problem does, however, state that the surveyors are "on the south bank" of the river. No doubt we can use the positions of the post & the surveyors as the positions of each bank of this river.

 

Your first picture should show an aerial view of the river - with two parallel lines (river banks), the post P on the north bank, and the surveyors A & B on the south bank. Your second drawing should show a side view - with surveyors A & B on opposite ends of the base of a triangle, and the post P rising up to the apex of the triangle.

 

I must ask for a clarification here: regarding the distance between the surveyors, you wrote "2/7a" - or 2 divided by 7a. This might or might not be accurate & can affect your answer. It seems more likely that it should be (2/7)a or 2a/7. I will assume it is (2/7)a, but you can solve this problem using whichever distance you believe is correct.

 

Now, fill in the information given in this problem. The two surveyors are (2/7)a apart in both drawings. In the aerial drawing, angle P (the post) is 150 degrees. In the side view, angle A is 45 degrees & angle B is 30 degrees.

 

Next, we establish variables for measurements we want to find. Angle Q (the top of the post) in the side view makes a triangle with the surveyors A & B - this will be an easy angle to measure. We primarily want to know the height of the post P and the width of the river, which we can call R. Let's also call the base of the post P, and the height of the post PQ.

 

The measure of angle Q is 105 degrees, because angle A + angle B equal 75 degrees. Subtract 75 from 180 to get the measure of angle Q.

 

In the side view, you might notice that the post is the same height as the distance from surveyor A to the post. How do I know this? The angle of elevation from surveyor A to the top of the post Q is 45 degrees. We assume that the post is erected at a perfect right angle to the ground, so that the post makes a perfect right angle with surveyor A. Using elimination, we find that 105 degrees (angle Q) minus 45 degrees (angle AQP) equals 60 degrees (angle BQP).

 

At this point, you should be able to use sine, cosine, and tangent techniques to find the measure of AP and BP. Once you have those, simply use them in the aerial view - the distances are the same. Finally, you will use sine, cosine, and tangent to deduce the width of the river.

 

I will check on your progress later & offer further guidance. Good luck!