Question

If the angle of elevation of a cloud from a point $h$ metres above a lake is $\alpha$, and the angle of depression of its reflection in the lake be $\beta$ and the distance of the cloud from the point of observation is $x$. Find $x$ if $\alpha ={30}^o,\beta ={45}^o$ and $h=250m$

• A. $1369m$
• B. $1672m$
• C. $1428m$
• D. $1267m$

Answer 1

The correct option is A $1369m$

Let a be a point $h$ metres above the lake $AF$ and $B$ be the position of the cloud.

Draw a line parallel to $EF$ from $A$ on $BD$ at $C.$

But,$BF=DF$

Let, $BC=m$

so, $BF=(m+h)$

$\Rightarrow BF=DF=(m+h)$

Consider $△BAC,$

$AB=m\text{cosec}\alpha ................\left(1\right)$

and, $AC=m\cot \alpha$

Consider $△ACD,$

$AC=(2h+m)\cot \beta$

Therefore, $m\cot \alpha =(2h+m)\cot \beta$

$\Rightarrow m=2h\cot \beta (\cot \alpha -\cot \beta )$

Substituting the value of $m$ in (1) we get,

$AB=\cos ec\alpha \left[2h\cot \beta /\right(\cot \alpha -\cot \beta \left)\right]=2hsec\alpha (\tan \beta -\tan \alpha )$

Now, Putting given values in above Equation

we get, $\frac{2\times 250\times \frac{2}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}=\frac{1000}{\sqrt{3}-1}=\frac{1000}{.73}=1369$

Therefore, Correct Answer is $1369$