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A person $X$ standing on a horizontal plane, observes a bird flying at a distance of $100 m$ from him at an angle of elevation of $30 °$ . Another person $Y$ standing on the roof of a $20$ m high building, observes the bird at the same time at an angle of elevation of $45 °$ . If $X$ and $Y$ are on the opposite sides of the bird, then find the distance of the bird from $Y$ .

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30 \sqrt{2} m

20 \sqrt{2} m

30 \sqrt{3} m

None of these

answer is A.

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Detailed Solution

It is given that the person

X

stands on a horizontal plane and observes a bird flying at a distance of

100 m

from him. The angle of elevation of the bird is

30 °

. Now another person

Y

stands on the roof of a

20

m high building and observes the bird at the same time at an angle of elevation of

45 °

.
According to the given data the figure has been drawn below.
Question Image

For

ΔXBA,

sin θ = P e r p e n d i c u l a r H y p o t e n u s e

\sin 30 ° = \frac{BA}{XB}

\Rightarrow \frac{1}{2} = \frac{BA}{100}

∵ sin 30 ° = 12

\Rightarrow \frac{100}{2} = BA

\Rightarrow 50 = BA

For

ΔBYD,

sin θ = P e r p e n d i c u l a r H y p o t e n u s e

\sin 45 ° = \frac{BD}{BY}

\Rightarrow \frac{1}{\sqrt{2}} = \frac{BA - AD}{BY}

∵ sin 45 ° = 12

\Rightarrow \frac{1}{\sqrt{2}} = \frac{BA - YC}{BY}

[as

AD = YC

]

\Rightarrow \frac{1}{\sqrt{2}} = \frac{50 - 20}{BY}

\Rightarrow \frac{1}{\sqrt{2}} = \frac{30}{BY}

\Rightarrow BY = 30 \sqrt{2}

∴

The distance of the bird from Y is

30 \sqrt{2} m

.
So the correct option is1.

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