A balloon is moving up in air vertically above a point \(A\) on the ground. When it is at a height \(h _{1},\) a girl standing at a distance \(d\) (point \(B\) ) from \(A\) (see figure) sees it at an angle \(45^{\circ}\) with respect to the vertical. When the balloon climbs up a further height \(h _{2},\) it is seen at an angle \(60^{\circ}\) with respect to the vertical if the girl moves further by a distance \(2.464\, d\) (point \(C\) ). Then the height \(h _{2}\) is (given tan \(\left.30^{\circ}=0.5774\right)\)\(.......\)

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Answer & Solution

Correct Answer

(A) \(d\)

Detailed explanation

\(\frac{ h _{1}}{ d }=\tan 45^{\circ} \Rightarrow h _{1}= d \ldots(1)\)

\(\frac{ h _{1}+ h _{2}}{ d +2.464 d }=\tan 30^{\circ}\)

\(\Rightarrow\left( h _{1}+ h _{2}\right) \times \sqrt{3}=3.46 d\)

\(\left(h_{1}+h_{2}\right)=\frac{3.46 d }{\sqrt{3}}\)

\(\Rightarrow d + h _{2}=\frac{3.46 d }{\sqrt{3}}\)

\(h _{2}= d\)

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