Question

38. Light house is a tower which is designed to exit light from a system of lamps and to

serve as a navigational aid for maritime pilots at sea. Two boats approaching a light
house in mid sea from opposite direction as shown in figure. The angle of elevation
of light house from both boats are also given.
45°
B(Boat)
(i) Find the height of the light house.
(ii) Find the distance BP.
(iii) Find the distance AB.

Answer 1

(i) height: 36.6 m

(ii) BP = 36.6 m

(iii) AB = 51.8 m

(iv) AC = 73.2 m

Explanation:

Given angles of elevation of 45° and 30° to the same point A from locations 100 m apart, you want to know the height of A, and distances BP, AB, and AC in the diagram.

Special triangles

The figure is divided into two special triangles. The one on the left is an isosceles right triangle with side lengths in the ratio 1:1:√2. The one on the right is a 30°-60°-90° triangle with side lengths in the ratio 1:√3:2. The common side AP corresponds to one ratio unit in each of these ratios.

(i) and (ii)

BP = AP = 1 ratio unit means BP : PC = 1 : √3, and ...

BP/BC = 1/(1+√3)

BP = BC/(1+√3) ≈ (100 m)/2.73205

BP ≈ 33.6 m

AP = 33.6 m . . . . . . same as BP

(iii)

AB is √2 times AP, so is ...

AB = √2 · 36.6 m

AB ≈ 51.8 m

AC

AC is 2 times AP, so is ...

AC = 2·(36.6 m)

AC = 73.2 m