Question

A man standing on the bank of a river observes that the angle subtended by a tree standing on the opposite bank is 60° on his side of Bank. When he moved away 24 m from the bank, he finds the angle to be 30°. Find the breadth of the river:

• A. 12
• B. 18
• C. 26
• D. 20

Answer 1

The Correct Option is A

To solve the problem of finding the breadth of the river, we apply trigonometric concepts involving right-angled triangles. The man first sees the tree at a 60° angle and later at a 30° angle after moving 24m away from the riverbank.
Let the breadth of the river be \(x\) meters and the height of the tree be \(h\) meters.
The first observation forms a right triangle with the riverbank, where the angle is 60°. Thus, using the tangent function:
\[ \tan(60^\circ) = \frac{h}{x} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{h}{x} \] (Equation 1)
After moving 24m away, the angle is now 30°. The distance from the man to the tree is now \(x + 24\). Again using tangent:
\[ \tan(30^\circ) = \frac{h}{x + 24} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x + 24} \] (Equation 2)
From Equation 1 and Equation 2, we equate the expressions for \(h\) and solve for \(x\). From Equation 1:
\[ h = x\sqrt{3} \]
Substitute \(h\) in Equation 2:
\[ \frac{1}{\sqrt{3}} = \frac{x\sqrt{3}}{x + 24} \]
Cross-multiply:
\[ x + 24 = 3x \]
Simplify:
\[ 2x = 24 \]
Thus:
\[ x = 12 \]
Hence, the breadth of the river is \(12\) meters.