Question

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 \(\tan 30^\circ = 0.5774\))

• A. \(0.464\,d\)
• B. \(d\)
• C. \(0.732\,d\)
• D. \(1.464\,d\)

Hint:

Angles are measured from the vertical, so horizontal distance = height \(\times \tan(\text{angle})\). Apply this at B and at C.

Answer 1

The Correct Option is B

The angle is measured from the vertical, so for a horizontal distance \(x\) and height \(h\): \(\tan\theta = x/h\).
At B: \(\tan 45^\circ = d/h_1 \Rightarrow h_1 = d\).
At C: horizontal distance from A is \(d + 2.464\,d = 3.464\,d\), height is \(h_1 + h_2\). So \(\tan 60^\circ = \dfrac{3.464\,d}{h_1 + h_2}\).
\(\tan 60^\circ = 1/\tan 30^\circ = 1/0.5774 = 1.732\).
\(h_1 + h_2 = \dfrac{3.464\,d}{1.732} = 2d\).
\(h_2 = 2d - d = d\).
Answer: (2) \(d\)