Question

A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is $60°$. When he retreats $20$ feet from the bank, he finds the angle to be $30°$. The breadth of the river in feet is

• A. $15$
• B. $15\sqrt{3}$
• C. $10\sqrt{3}$
• D. $10$

Answer 1

The correct option is D

$10$

Explanation for the correct option:

Step 1: Find the tangent relation between the height of the tree and the breadth of the river:

Assume the breadth of the river be $x$ and the height of the tree as $y$.

Apply tangent relation for $60°$ angle.

$$\begin{gathered} \Rightarrow \tan 60°=\frac{h}{x} \\ \Rightarrow \sqrt{3}=\frac{h}{x}.......\left[1\right] \end{gathered}$$

Step 2: Find the tangent relation between the height of the tree and the breadth of the river and $20$ m distance:

Use the base as $x+20$ and the height of the tree as $y$.

Apply tangent relation for $30°$ angle.

$$\begin{gathered} \Rightarrow \tan 30°=\frac{h}{x+20} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+20}.......\left[2\right] \end{gathered}$$

Step 3: Divide both the equations:

Divide equation $\left[1\right]$ with $\left[2\right]$ and eliminate $h$.

$$\begin{gathered} \Rightarrow \frac{x+20}{x}=3 \\ \Rightarrow 3x=x+20 \\ \Rightarrow x=10 \end{gathered}$$

Hence, option (Dis the correct answer.