Question : Let A and B be two towers with the same base. From the midpoint of the line joining their feet, the angles of elevation of the tops of a and b are 30o and 45o, respectively. The ratio of the heights of A and B is:
Option 1: $1: 3$
Option 2: $1: \sqrt{3}$
Option 3: $\sqrt{3}: 1$
Option 4: $3: 1$
Correct Answer: $1: \sqrt{3}$
Solution :
In $\triangle ACE$,
$\frac{AC}{CE} = \tan 30^{\circ}$
⇒ $\frac{AC}{CE} = \frac{1}{\sqrt3}$
⇒ $CE = \sqrt3 AC$ -------------(i)
In $\triangle BDE$,
$\frac{BD}{ED} = \tan 45^{\circ}$
⇒ $\frac{BD}{ED} = 1$
⇒ $ED = BD$ -------------(ii)
Since CE = ED,
⇒ $\sqrt 3 AC = BD$
⇒ $\frac{AC}{BD} = \frac{1}{\sqrt3}$
So, the ratio of heights = $1: \sqrt3$
Hence, the correct answer is $1: \sqrt3$.
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