A balloon is moving up in air vertically above a point A on the ground. When it is at a height h₁, a girl standing at a distanced (point B) from A (see figure) sees it at an angle 45ᵒ with respect to the vertical. When the balloon climbs up a further height h₂, it is seen at an angle 60ᵒ with respect to the vertical if the girl moves further by a distance 2.464 d(point C). Then the height h₂ is (given tan 30ᵒ = 0.5774)

Solution

Physics sir · 4 steps

The angles are from the vertical, so tan of each angle is ground distance divided by height.

1. Read the angle. Each angle is measured from the vertical. So in each right triangle, tan θ = (horizontal distance from A)/(height of balloon).
2. First sighting. From B the horizontal distance is d and the angle is 45°. So tan 45° = d/h₁, which gives h₁ = d.
3. Second sighting. C is at a distance AC = d + 2.464d = 3.464d from A, and the balloon is now at height h₁ + h₂. The angle is 60°, and tan 60° = 1/(tan 30°).

   h₁ + h₂ = 3.464d/(tan 60°) = 3.464d × tan 30° = 3.464d × 0.5774 ≈ 2d

4. So. h₂ = 2d - h₁ = 2d - d = d, which is option (b).

The idea to keep. Always check which line the angle is measured from. From the vertical, tan is horizontal over vertical. Measure every distance from the foot of the height, not from the last point.

Answer from the NTA final answer key. Solution by PrepWiser.