Question

If two towers of heights h1 and h2 subtend angles of 45° and 30° respectively at the midpoint of the line joining their feet, then the ratio of h₁: h₂ is

• A. \(1:\sqrt3\)
• B. \(\sqrt3:1\)
• C. 1:3
• D. 3:1

Answer 1

The Correct Option is B

Let the distance between the towers' feet be \( d \). We are given that the angle of elevation from the midpoint of the line joining their feet to the top of the first tower is \( 45^\circ \), and to the second tower is \( 30^\circ \). The height of the towers are \( h_1 \) and \( h_2 \) respectively.
The tangent of an angle in a right-angled triangle is given by the ratio of the opposite side to the adjacent side. Therefore, for the first tower: \[ \tan 45^\circ = \frac{h_1}{\frac{d}{2}} = 1 \quad \Rightarrow \quad h_1 = \frac{d}{2} \] For the second tower: \[ \tan 30^\circ = \frac{h_2}{\frac{d}{2}} = \frac{1}{\sqrt{3}} \quad \Rightarrow \quad h_2 = \frac{d}{2\sqrt{3}} \] Now, the ratio of the heights \( h_1 : h_2 \) is: \[ \frac{h_1}{h_2} = \frac{\frac{d}{2}}{\frac{d}{2\sqrt{3}}} = \sqrt{3} : 1 \]

The correct option is (B): \(\sqrt3:1\)