Question

If two towers of height $h_{1}$ and $h_{2}$ subtend angles $45^{\circ}$ and $30^{\circ}$ respectively at the midpoint of the line joining their feet, then find the ratio of $h_{1}$ to $h_{2}$: A) $1$ to $2$ B) $3$ to $2$ C) $\sqrt{3}$ to $1$ D) $1$ to $\sqrt{3}$

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Answer

The correct option is C: $\sqrt{3}: 1$

Let $h_{1}$ and $h_{2}$ denote the heights of the two towers, and let $2x$ be the total distance between them.

Using trigonometric relationships at the midpoint of the line joining their feet, we have:

  1. For the tower with height $h_{1}$: $$ \tan 45^{\circ} = \frac{h_{1}}{x} $$ Since $\tan 45^{\circ} = 1$, this simplifies to: $$ 1 = \frac{h_{1}}{x} \quad \Rightarrow \quad h_{1} = x $$

  2. For the tower with height $h_{2}$: $$ \tan 30^{\circ} = \frac{h_{2}}{x} $$ Since $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$, this simplifies to: $$ \frac{1}{\sqrt{3}} = \frac{h_{2}}{x} \quad \Rightarrow \quad h_{2} = \frac{x}{\sqrt{3}} $$

Dividing $h_{1}$ by $h_{2}$ gives: $$ \frac{h_{1}}{h_{2}} = \frac{x}{\frac{x}{\sqrt{3}}} = \sqrt{3} $$

Therefore, the ratio $$ h_{1} : h_{2} $$ is $\sqrt{3}: 1$, confirming the correct option as C: $\sqrt{3}: 1$.


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