Question : From the peak of a hill 300 m high, the angle of depression of two sides of a bridge lying on the ground are $45°$ and $30°$ (both ends of the bridge are on the same side of the hill). Then the length of the bridge is:
Option 1: $300(\sqrt3 - 1)$ m
Option 2: $300(\sqrt3 + 1)$ m
Option 3: $300\sqrt3$ m
Option 4: $\frac{300}{\sqrt3}$ m
Correct Answer: $300(\sqrt3 - 1)$ m
Solution : $AB$ = height of peak = 300 m $CD$ = length of Bridge We are given the angle of depression, i.e., $\angle EAC = 45°$ and $\angle EAD = 30°$ ⇒ $\angle ACB = 45°$ and $\angle ADB = 30°$ Now, \(In \triangle ABC\) \(\tan\theta = \frac{P}{B}\) \(\tan45 ^{\circ} =\frac{AB}{BC}\) \(1=\frac{AB}{BC}=AB:BC=1:1\) \(In\ \triangle ABD,\) ⇒ \(\tan30 ^{\circ} =\frac{AB}{BD}\) ⇒ \(\frac{1}{\sqrt{3}}=\frac{AB}{BD}\Rightarrow AB:BD=1:\sqrt{3}\) Now, \(CD = BD - BC\) ⇒ \(CD \Rightarrow \sqrt{3}-1\) $AB$ = 1 unit = 300 metre So, $CD=(\sqrt{3}-1)$ units = $300(\sqrt{3}-1)$ metres Hence, the correct answer is $300(\sqrt3 - 1)$ m.
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