If the angle of elevation of a cloud from a point h metres above the lake be 30° and the angle of depression of its reflection in the lake be 60°, prove that the height of the cloud is 2h.
Answers
Answer:
Thus, the height of the opposite house is h(1+tan cot) metres.
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Step-by-step explanation:
the angle of elevation of a cloud from point h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the distance of the cloud from the point of observation is
tanβ−tanα
2hsecα
ANSWER
Let AB be the surface of the lake and let C be a point of observation such that AC = h metres Let D be the position of the cloud and D be its reflection in the lake Then BD = BD
In Δ DCE
tanα=
CE
DE
⇒CE=
tanα
H
............(i)
In Δ CED'
⇒CE=
tanβ
h+H+h
⇒CE=
tanβ
2h+H
............(ii)
From (i) & (ii)
⇒
tanα
H
=
tanβ
2h+H
⇒Htanβ=2htanα+Htanα
Htanβ−Htanα=2htanα⇒H(tanβ−tanα)=2htanα
⇒H=
tanβ−tanα
2htanα
.........(iii)
In Δ DCE sinα=
CD
DE
⇒CD=
sinα
DE
⇒⇒CD=
sinα
H
Substituting the value of H from (iii)
CD=
(tanβ−tanα)sinα
2htanα
⇒CD=
(tanβ−tanα)sinα
2h
cosα
sinα
CD=
tanβ−tanα
2hsecα
Hence the distance of the cloud from the point of observation is
tanβ−tanα
2hsecα
solution