Question

Two boys on either side of a temple of 45 meters height observe its top at the angles of elevation 30° and 60° respectively. Find the distance between the two boys.

• A. \(60\sqrt{3}\) m
• B. \(40\sqrt{3}\) m
• C. \(60/\sqrt{3}\) m
• D. \(40/\sqrt{3}\) m

Hint:

When dealing with angles of elevation and height, use the tangent function to relate the height and distance from the base.

Answer 1

The Correct Option is B

Let the distance between the two boys be \(d\). We will use trigonometric functions to find this distance. The height of the temple is \(45 \, \text{m}\), and the angles of elevation are \(30^\circ\) and \(60^\circ\). Let the distances from the temple to the boys be \(x_1\) and \(x_2\) respectively. Using the tangent function: \[ \tan(30^\circ) = \frac{45}{x_1} \quad \text{and} \quad \tan(60^\circ) = \frac{45}{x_2} \] We know that \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) and \(\tan 60^\circ = \sqrt{3}\). Thus: \[ \frac{45}{x_1} = \frac{1}{\sqrt{3}} \quad \Rightarrow \quad x_1 = 45\sqrt{3} \] \[ \frac{45}{x_2} = \sqrt{3} \quad \Rightarrow \quad x_2 = 15/\sqrt{3} \] The total distance between the two boys is: \[ d = x_1 + x_2 = 45\sqrt{3} + 15/\sqrt{3} = 40\sqrt{3} \, \text{m} \] Thus, the correct answer is option (2).