Question

The base of a cliff is circular. From the extremities of a diameter of the base angles of elevation of the top of the cliff are ${30}^{\circ }$ and $60°$. If the height of the cliff is $500$ m, then the diameter of the base of the cliff is

• A. $\frac{2000}{\sqrt{3}}m$
• B. $\frac{1000}{\sqrt{3}}m$
• C. $\frac{2000}{\sqrt{2}}m$
• D. $\frac{1000}{\sqrt{2}}m$

Answer 1

The correct option is A

$\frac{2000}{\sqrt{3}}m$

Explanation for the correct option:

Finding the diameter:

It is known that $\tan \theta =\frac{\mathrm{Perpendicular}}{\mathrm{base}}$

Put $\theta =60°$ and perpendicular length is $500$.

So,

$$\begin{gathered} \tan 60°=\frac{CE}{AE} \\ \Rightarrow \sqrt{3}=\frac{500}{AE} \\ \Rightarrow AE=\frac{500}{\sqrt{3}} \end{gathered}$$

Similarly, putting $\theta =30°$, we get

$$\begin{gathered} \tan 30°=\frac{CE}{BE} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{500}{BE} \\ \Rightarrow BE=500\sqrt{3} \end{gathered}$$

$$\begin{gathered} \mathrm{diameter}=AE+BE \\ =\frac{500}{\sqrt{3}}+500\sqrt{3} \\ =\frac{500+500\left(3\right)}{\sqrt{3}} \\ =\frac{500+1500}{\sqrt{3}} \\ =\frac{2000}{\sqrt{3}} \end{gathered}$$

Therefore, the correct answer is option (A).