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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