Question

The angles of elevation of top of a lighthouse from 3 boats A ,B and C in a straight line of the same side of lighthouse are a , 2 a , 3 a respectively .If the distance between the boats A and B and thd boats B and C are x and y respectively. find the height of lighthouse .

Answer 1

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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

h(tan2a - tana) = xtana.tan2a

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Answer 2

$\huge{ \bf{ \red{answer}}}$

★ height of lighthouse = x sin 2 a

$\huge{ \bf{ \red{important \: formulae}}}$

$\bullet\sf{tan(x + y) = \frac{tan \:x + tan \: y}{1 - tan \: x \: tan \: y} }$

$\bullet \sf{2 \: sin x \: cos x = sin \: 2 \: x}$

( where x and y are denoting angles )

$\huge{ \bf{ \red{solution}}}$

➹ Refer to the attachment for Solution .