Trigonometry-20

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Two boats approach a light house in mid sea from opposite directions. The angles of elevation of

the top of the light house from the two boats are 30⁰ and 45⁰ respectively. If the distance between

the two boats is 100 meters, find the height of the light house.

Solution: S and S' are ships. L.T. is the light house. SS' = 100 m

Let LT = x m

In the right angled isosceles triangle LTS',

LS' = LT = x m

Therefore, SL = (100 - x) m

In the triangle LTS, TL/SL = tan 30⁰

i.e. x / (100 - x) = 1/√3

(100 - x) = x √3

x + x √3 = 100

x (1 + √3) = 100

x = 100/(1 + √3) = 100/(1 + √3) x (1 - √3)/(1 - √3)

= 100 (1 - √3) / ( 1-3) = 100 (1 - √3)/-2 = 100(√3 - 1)/2

Hence, the height of the light house = 50 (√3 - 1) m = 36.6 m