Question : The two banks of a canal are straight and parallel. A, B, and C are three persons, of whom A stands on one bank and B and C on the opposite banks. B finds the angle ABC is 30°, while C finds that the angle ACB is 60°. If B and C are 100 metres apart, the breadth of the canal is:
Option 1: $\frac{25}{\sqrt{3}}$ metres
Option 2: $20\sqrt{3}$ metres
Option 3: $25\sqrt{3}$ metres
Option 4: $\frac{20}{\sqrt{3}}$ metres
Correct Answer: $25\sqrt{3}$ metres
Solution :
BD = $x$ metre (let)
∴ CD = $(100 - x)$ metres
AD ⊥ BC; AD = $y$ metres
From ∆ ABD,
tan 30° = $\frac{AD}{BD}$
$⇒\frac{1}{\sqrt{3}}=\frac{y}{x}$
$\therefore x=\sqrt{3}y$ ----(1)
From ∆ACD,
$\tan 60° = \frac{AD}{CD}$
$⇒\sqrt{3}=\frac{y}{(100 - x)}$
$\therefore y=(100 - x)\sqrt{3}$
Using the value of $x$ from equation 1, we get,
$⇒y=(100-\sqrt{3}y)\sqrt{3}$
$⇒4y=100\sqrt{3}$
$\therefore y= 25\sqrt{3}$ metres
Hence, the breadth of the canal is $25\sqrt{3}$ metres.
Hence, the correct answer is $25\sqrt{3}$ metres.
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