Discussion
A man observes the elevation of a balloon to be \( 45^{\circ} \) at a point A. He then walks towards the balloon and at a point B finds the elevation to be \( 60^{\circ} \). Walking further in the same direction he finds the balloon directly overhead at a height of 450 m. The distance travelled from A to B is
Answer: A
The balloon ends up directly overhead at 450 m, so measure both earlier observations from that vertical line.
From A, at \( 45^{\circ} \):
\[ d_{A} = \frac{450}{\tan 45^{\circ}} = 450 \text{ m} \]
From B, at \( 60^{\circ} \):
\[ d_{B} = \frac{450}{\tan 60^{\circ}} = \frac{450}{\sqrt{3}} = 150\sqrt{3} = 259.8 \text{ m} \]
\[ AB = d_{A} - d_{B} = 450 - 150\sqrt{3} = 150\left(3 - \sqrt{3}\right) \]
\[ = 150 \times 1.268 = 190.2 \text{ m} \]
The balloon is assumed to hold its height while the man walks, so all three sightings share the same 450 m - that is what makes the two distances comparable.
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