Question

The angle of elevation of a cloud from a point $200$ metres above a lake is ${30}^o$ and the angle of depression of the reflection of the cloud in the lake is ${60}^o$. Find the height of the cloud.

• A. The height of the cloud$=400m$
• B. The height of the cloud$=280m$
• C. The height of the cloud$=340m$
• D. None of these

Answer 1

Answer: (A)

The height of the cloud$=400m$

Let $QR$ be the surface of the lake and $P$ be the point of observation which is $200m$ above $Q$
i.e., $PQ=200m$. Let $C$ be the position of the cloud.
Through $C$ draw $CR$ perpendicular to th esurface of the lake and $D$ the reflection of the cloud in the lake then $CR=RD=h$ metres. (Assume) (laws of relection)
Through $P$ draw $PQ\perp$ on the surface of lake and $PM\perp CD$.
$\angle CPM={30}^o,\angle MPD={60}^o$
$MR=PQ=200m$, Let $PM=x$ metres
$CM=CR-MR=(h-200m)$, ($\because CR=h$ metres)
$DM=DR+RM=(h+200m)$, ($\because DR=h$ metres)
In rt. $△CMP$
$\frac{CM}{PM}=\tan {30}^o$
$\frac{h-200}{x}=\frac{1}{\sqrt{3}}$
$x=\sqrt{3}(h-200)........\left(1\right)$
From rt. $△PMD$,
$\frac{MD}{PM}=\tan {60}^o$
$\frac{h+200}{x}=\sqrt{3}$
$x=\frac{h+200}{\sqrt{3}}........\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$, we get
$\sqrt{(}h-200)=\frac{h+200}{\sqrt{3}}$
$\Rightarrow$ $3(h-200)=h+200$
$\Rightarrow$ $3h-h=600+200=800$
$\Rightarrow$ $2h=800$
$\Rightarrow$ $h=\frac{800}{2}=400$
Required height of the cloud$=h=CR=400m$