Two men on either side of the cliff 80 meters high observe the angles of elevation of the top of the cliff to be $30{}^\circ $ and $60{}^\circ $ , respectively. Find the distance between the two men.

Question

Vedantu Content Team · Accepted Answer

Hint: Assume that the height of the cliff from the ground is ‘h’. First, draw a rough diagram of the given conditions and then use the formula $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$ in the different right angle triangles and substitute the given values to get the height.

Complete step-by-step answer:

Let us start with the question by drawing a representative diagram of the situation given in the question.

According to the above figure:

We have assumed the height of the tower from the ground as ‘h’. Therefore, AB = h. Also, assume that the distance BD is ‘x’ meters and distance BE is ‘y’ meters.

Now, in right angle triangle ADB,

$\angle ADB=30{}^\circ $

We know that, $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$. Therefore,

\[ \tan 30{}^\circ =\dfrac{AB}{BD} \]

$ \Rightarrow \tan 30{}^\circ =\dfrac{h}{x} $

$ \Rightarrow x=\dfrac{h}{\tan 30{}^\circ } $

And we know that the value of $\tan 30{}^\circ $ is equal to $\dfrac{1}{\sqrt{3}}$ .

$\therefore x=\sqrt{3}h.............(i)$

Now, in right angle triangle AEB,

\[\angle AEB=60{}^\circ \]

We know that, $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$. Therefore,

\[\tan 60{}^\circ =\dfrac{AB}{BE} \]

$ \Rightarrow \tan 60{}^\circ =\dfrac{h}{y} $

$ \Rightarrow y=\dfrac{h}{\tan 60{}^\circ } $

And we know that the value of $\tan 60{}^\circ $ is equal to $\sqrt{3}$ .

$\therefore y=\dfrac{h}{\sqrt{3}}.............(ii)$

Now according to the data given in the question:

The distance between both men=BE+BD

$\Rightarrow y+x=\text{ distance between two men}$

Now we will substitute x and y from equation (i) and equation (ii), respectively. On doing so, we get

$\sqrt{3}h+\dfrac{h}{\sqrt{3}}=\text{ distance between two men}$

Now from the question, we know that the value of h is 80 meters.

$\therefore \text{distance between both men=80}\sqrt{3}+\dfrac{80}{\sqrt{3}}$

$\Rightarrow \text{distance between both men=}\dfrac{320}{\sqrt{3}}\text{ meters}$

So, the distance between two men is $\dfrac{320}{\sqrt{3}}$ meters.

Note: Do not use any other trigonometric function like sine or cosine of the given angle because the information which is provided to us is related to the height of the triangle, and we have to find the base. So, the length of the hypotenuse is of no use. Therefore, the formula of the tangent of the angle is used. We can use sine or cosine of the given angles but then the process of finding the height will be lengthy.