(Q2) The angles of elevation of the top of a lighthouse from 3 boats A, B and C in a straight line of same side of the light- house are a, 2a, 3a respectively. If the distance between the boats A and B is x meters. Find the height of lighthouse.

Solution :

APQBCxyxa2a3ah

From the figure,

Let PQ be the height of the lighthouse = h m

A = First point of observation

B = Second point of observation

C = Third point of observation Given,

AB = x and BC = y

(Not given in the text)

Exterior angle = Sum of the opposite interior angles

∠PBQ = ∠BQA + ∠BAQ and

∠PCQ = ∠CBQ + ∠CQB

∴ AB = X = QB

By applying the sine rule,

∆BQC we get
BQ
sin∠QCB
 =
BC
sin∠CQB
x
sin(180° - 3a)
 =
y
sin a
x
y
 =
sin 3a
sin a
 =
sin a - sin3a
sin a
 = - sin2a
sin2a = -
x
y
 =
y - x
y
sin2a =
y - x
y
∴cos2a = 1 - sin2a = 1 -
y - x
y
 =
y - y + x
y
 =
y + x
y

From ∆PBQ

sin2a =
h
x
2sin a cos a =
h
x
4sin2a cos2a =
h2
x2

Put sin2a value and cos2a value

4.
y - x
y
 .
x + y
y
 =
h2
x2
h2 =
x2
y2
 (y - x)(x + y)
∴h =
x
y
 √[(y - x)(x + y)]
∴ Height of lighthouse =
x
y
 √[(y - x)(x + y)] meters