Question

the angle of elevation of a cloud from a point 80m above a lake 30° and the angle of depression of the reflection of cloud in the lake is 60 find the height of the cloud​

Answer 1

Appropriate Question :-

The angle of elevation of a cloud from a point 80m above the lake level is 30° and the angle of depression of the reflection of cloud in the lake is 60°. Find the height of the cloud.

$\large\underline{\sf{Solution-}}$

Let assume that XY be the surface of lake level and let C be the point of observation which is at a height 25 m above the lake level.

Let further assume that D be the position of stationary cloud and E be its reflection in the lake.

Construction :- Join DE intersecting the surface XY at B. From C, Draw CF perpendicular to DE.

Let assume that height of the cloud, DB = h m

Now, as lake surface act as a plane mirror.

So, DB = BE = h m

Let further assume that CF = x m

Now, In right angle triangle DFC

$\rm \: tan30\degree = \dfrac{DF}{FC}$

$\rm \: \dfrac{1}{ \sqrt{3} } = \dfrac{h - 80}{x}$

$\rm\implies \:x \: = \: \sqrt{3}(h - 80) - - - (1)$

Now, In right angle triangle CFE

$\rm \: tan60\degree = \dfrac{FE}{FC}$

$\rm \: \sqrt{3} = \dfrac{80 + h}{x}$

On substituting the value of x, from equation (1), we get

$\rm \: \sqrt{3} = \dfrac{80 + h}{ \sqrt{3}(h - 80) }$

$\rm \: 80 + h = 3(h - 80)$

$\rm \: 80 + h = 3h - 240$

$\rm \: 3h - h = 240 + 80$

$\rm \: 2h= 320$

$\rm \: h= 160 \: m \:$

Hence, the height of the cloud is 160 m

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$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$