Question

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 light house?

Answer 1

HELLO DEAR,

[figure is in the attachment ]

here, <Q = 90, PQ = h , AB = x
So, IN ∆ BPQ,

tan2a = PQ/QB

QB = h/tan2a----------( 1 )

now, IN ∆PQC,

tan3a = PQ/QC

QC = h/tan3a-----------( 2 )

similarly, IN ∆APQ

tana = PQ/(QA)

tana = h/(QC + BC + AB) [ as QA = QC + BC + AB]

(QC + BC + x) = h/tana [ AB = x]----------( 3 )



We may write [BC = QB - QC] we get,

[ QC + QB - QC + X] = h/tana

From-------( 1 ) , & -------( 3 ).

[h/tan 2a + x] = h/tana
​​
[(h + xtan2a)/tan2a] = h/tana

tana[h + xtan2a] = htan2a

htana + xtana.tan2a = htan2a

h(tan2a - tana) = xtana.tan2a

$\sf{\large{\boxed{\bold{h} = \bold{\frac{x.tana.tan2a}{tan2a - tana}}}}}$.


I HOPE ITS HELP YOU DEAR,
THANKS