Rational Trigonometry/11

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Applications of rational trig to surveying (I)

Mountain Problem
Suppose there is some mountain of height h. From far away, we measure the spread from the ground to the top as S1=0.182. We then proceed closer towards the mountain, so quadrance Q3=1300 2 . We then measure a new spread to the top, S2=0.328. There is now a triangle formed between all these measurements to the top of the mountain.

The spread at the top of the mountain is some S3, which we would assume can be solved using the Tripe Spread Formula. First, however, it might be helpful to rewrite the Triple Spread Formula by solving for S3, like such:

$$ (s_3-(s_1+s_2-2s_1s_2))^2 = 4s_1s_2(1-s_1)(1-s_2) $$

(It is highly recommended you practice deriving this formula from the given form of the Triple Spread Formula yourself!) As a quadratic, this offers us two solution for S3, 0.751 or 0.028. Clearly, the small choice is indicated. From here, the Spread Law tells us Q1. This leads immediately to h.