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 mou
Answer: C
Put the mountain's foot at the origin. The summit is seen at \( 45^{\circ} \) from there, so for a height \( H \) km the summit sits at \( (H,\;H) \).
Climbing 2 km up a \( 30^{\circ} \) incline moves the observer to
\[ \left(2\cos 30^{\circ},\; 2\sin 30^{\circ}\right) = \left(\sqrt{3},\; 1\right) \]
From there the elevation is \( 60^{\circ} \):
\[ \frac{H - 1}{H - \sqrt{3}} = \tan 60^{\circ} = \sqrt{3} \]
\[ H - 1 = \sqrt{3}H - 3 \quad\Longrightarrow\quad H(\sqrt{3} - 1) = 2 \]
\[ H = \frac{2}{\sqrt{3}-1} = \sqrt{3} + 1 = 2.73 \text{ km} \]
The ascent lifts the observer as well as moving them closer, so both coordinates change - treating the 2 km as horizontal only is the standard error.
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