Discussion
The elevation of the summit of a mountain from its foot is 45°. After ascending 2 km towards the mountain upon an incline of 30°,the elevation changes to 60°. What is the approximate height of the mountain?
Answer: B
Put the foot of the mountain at the origin. From there the summit is at 45°, so if the mountain is \( H \) km high the summit is at \( (H,\;H) \).
Walking 2 km up a 30° incline moves the observer to
\[ \left(2\cos 30^{\circ},\; 2\sin 30^{\circ}\right) = \left(\sqrt{3},\; 1\right) \]
From this new point the elevation is 60°:
\[ \tan 60^{\circ} = \frac{H - 1}{H - \sqrt{3}} = \sqrt{3} \]
\[ H - 1 = \sqrt{3}H - 3 \quad\Longrightarrow\quad H(1 - \sqrt{3}) = -2 \]
\[ H = \frac{2}{\sqrt{3} - 1} = \frac{2(\sqrt{3}+1)}{2} = \sqrt{3} + 1 = 2.73 \text{ km} \]
The ascent raises the observer as well as bringing them closer, so both coordinates must change - treating the 2 km as purely horizontal is the usual mistake.
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