Theory question 103
TrigonometryA simple measuring device is used at points X and Y on the same horizontal level to measure the angles of elevation of the peak P of a certain mountain. If X is known to be 5,200m above sea level, |XY| = 4,000m and the measurements of the angles of elevation of P at X and Y are 15° and 35° respectively, find the height of the mountain. (Take tan 15° = 0.3 and tan 35° = 0.7)
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Let h be the height of P above the level of X and Y, and O the point on that level directly below P. X, Y and O are in a line, Y nearer the mountain.
XO = h/tan 15° = h/0.3; YO = h/tan 35° = h/0.7
XO − YO = 4000: h(1/0.3 − 1/0.7) = 4000, h × 40/21 = 4000
h = 2100 m
Height of mountain above sea level = 5200 + 2100 = 7300 m
(If X and Y are taken on opposite sides of the mountain, XO + YO = 4000 gives h(1/0.3 + 1/0.7) = 4000, h = 840 m and the height is 6040 m.)
Express both horizontal distances in terms of h, then use the 4,000 m between X and Y.