A railway company is building a new railroad. Surveyors wish to measure the height of a hill and the length of a tunnel to be cut through the base of the hill. At the East end of the tunnel, the surveyors measure the angle of elevation of the top of the hill to be 60°. When they move 80 metres further East, the angle of elevation to the top of the hill is 45°. From a point 120 metres to the West of the West end of the tunnel, the angle of elevation of the top of the hill is 30°. Find the length of the tunnel

Question

A railway company is building a new railroad. Surveyors wish to measure the height of a hill and the length of a tunnel to be cut through the base of the hill. At the East end of the tunnel, the surveyors measure the angle of elevation of the top of the hill to be 60°. When they move 80 metres further East, the angle of elevation to the top of the hill is 45°. From a point 120 metres to the West of the West end of the tunnel, the angle of elevation of the top of the hill is 30°.  Find the length of the tunnel

Ms Pi 3.14159265358979323 · Answer

What an awesome question : ) I would personally start with a well drawn diagram. Then start at the place that has the most information. Note: you will have to do a couple of things then add up all the bits to find the length of the tunnel. I started at the EAST most side of the tunnel. You have a triangle furthest to the right of your picture. One angle is 45 degrees. The other base angle is found by supplementary angles (they share the straight line) so the one you don't need is the 60, so its supplement is 120. So your triangle has 45 degrees, 120 degrees and another angle to add up to 180 degrees. The missing angle must be 15 degrees. Then my first step is SINE LAW to find the leftmost side of the triangle we are working on : ) Again, awesome question !!