A ladder 10 m long reaches a point 10 m below the top of a vertical flagstaff. From foot of the ladder, the elevation of top of the flagstaff is 60°. What is the height of flagstaff?

Explanation

Let the ladder reach height h, making the flagstaff's total height h+10. The base distance x = \sqrt{10^2 - h^2}. The angle of elevation to the top is 60^{\circ}, so \tan 60^{\circ} = \frac{h+10}{x} \implies \sqrt{3} = \frac{h+10}{\sqrt{100-h^2}}. Squaring gives 3(100-h^2) = h^2+20h+100 \implies h^2+5h-50=0 \implies h=5. Total height is 5+10=15\text{ m}.

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