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Question

The angle of elevation of a cloud from a point 60 m above a lake is 30 and the angle of depression of the reflection of cloud in the lake is 60. Find the height of the cloud.  [4 MARKS]


Solution

Let AB be the surface of the lake and let P be a point vertically above A such that AP = 60 m



Let C be the position of the cloud and let D be its reflection in the lake.

Draw PQCD, Then,

QPC=30,QPD=60,

BQ = AP = 60 m

Let CQ=x metres. Then,

BD=BC=(x+60) m

From right ΔPQC, we have

PQCQ=cot30=3

PQx m=3PQ=x3 m  -----------(i)

From right ΔPQD, we have

PQQD=cot60=13

PQ(x+60+60)m=13PQ=(x+120)3 m  -------(ii)

Equating the values of PQ from (i) and (ii), we get

x3=(x+120)3 m

3x=x+1202x=120x=60

height of the cloud from the surface of the lake

=BC=(60+x)m=(60+60) m=120 m

Hence, the height of the cloud from the surface of the lake is 120 metres.

Alernative Method,

                                                                                                                         [12 MARK]

In ΔABE

tan 30=hx

13=hxx=h3(1)                            [1 MARK]

In ΔBDE

tan 60=120+hx

3=120+hh3

3h=120+h

2h=120

h=60 m                                                              [1 MARK]

Hence, height of cloud above the lake = 60 + h

                                                               = 60 + 60

                                                               = 120 m            [12 MARK]

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