Question

A person standing on the bank of the river observes that the angle of elevation of the top of a tree standing on the opposite bank is $60$ degrees. When he was 40m away from the bank he found the angle of elevation to be $30$ degrees. Find the width of the river.

• A. $20$ m
• B. $10$ m
• C. $5$ m
• D. $1$ m

Answer 1

The correct option is A

$20$ m

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.

$$\begin{gathered} \Rightarrow \tan 30°=\frac{h}{x+40} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+40}.......\left[2\right] \end{gathered}$$

Step 3: Divide both the equations:

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

$$\begin{gathered} \Rightarrow \frac{x+40}{x}=3 \\ \Rightarrow 3x=x+40 \\ \Rightarrow x=20 \end{gathered}$$

Hence, option (A) is the correct answer.