Discussion
At the foot of a mountain the elevation of its summit is 45º; after ascending 1000 m towards the mountain up a slope of 30º inclination is found to be 60º. Find the height of the mountain.
Answer: A
Place the mountain's foot at the origin. The summit is seen at \( 45^{\circ} \), so for a height \( H \) km it sits at \( (H,\;H) \).
Ascending 1000 m = 1 km up a \( 30^{\circ} \) slope moves the observer to
\[ \left(\cos 30^{\circ},\; \sin 30^{\circ}\right) = \left(\frac{\sqrt{3}}{2},\; \frac{1}{2}\right) \]
From there the elevation is \( 60^{\circ} \):
\[ \frac{H - \tfrac{1}{2}}{H - \tfrac{\sqrt{3}}{2}} = \sqrt{3} \]
\[ H - \frac{1}{2} = \sqrt{3}H - \frac{3}{2} \quad\Longrightarrow\quad H(\sqrt{3}-1) = 1 \]
\[ H = \frac{1}{\sqrt{3}-1} = \frac{\sqrt{3}+1}{2} = 1.366 \text{ km} \]
Climbing a slope raises the observer as well as bringing them nearer, so both coordinates change.
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