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If from the top of a light house $100 m$ high, the angles of depression made by two boats situated in a horizontal line along the foot of the light house are $45 °$ and $30 °$ . The distance between the two boats

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100 (\sqrt{3} - 1) m

100 (\sqrt{3} + 1) m

80 (\sqrt{3} - 1) m

None of these

answer is A.

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Detailed Solution

In $Δ A C D, \tan 45 ° = \frac{100}{A C}$

$\Rightarrow 1 = \frac{100}{A C}$

$\Rightarrow A C = 100 m$

In $∆ B C D, \tan 30 ° = \frac{C D}{B C}$

$\frac{1}{\sqrt{3}} = \frac{100}{B C}$

$\Rightarrow B C = 100 \sqrt{3}$

$A C + A B = 100 \sqrt{3}$

$100 + A B = 100 \sqrt{3}$

$A B = 100 (\sqrt{3} - 1) m$

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