From the top of a tower, the angle of depression to a point A on ground is \(30^\circ\). From the same point on the tower, the angle of elevation to a balloon vertically above point A is \(60^\circ\). If the horizontal distance between the tower and point A is 40 meter, the height (in meter) of the balloon above the point A is
\(\dfrac{80}{\sqrt{3}}\)
Let the height of the tower be \(h\) metres and the height of the balloon above the ground be \(H\) metres. The horizontal distance from the tower to point A is 40 m.
The angle of depression from the tower top to point A is \(30^\circ\), so \(\tan 30^\circ = \dfrac{h}{40}\), giving \(h = \dfrac{40}{\sqrt{3}}\).
The angle of elevation from the tower top to the balloon (which is vertically above A, at horizontal distance 40 m) is \(60^\circ\), so \(\tan 60^\circ = \dfrac{H - h}{40}\), giving \(H - h = 40\sqrt{3}\).
Adding: \(H = \dfrac{40}{\sqrt{3}} + 40\sqrt{3} = \dfrac{40 + 120}{\sqrt{3}} = \dfrac{160}{\sqrt{3}}\).
The height of the balloon above point A equals its height above the ground, which is \(\dfrac{160}{\sqrt{3}}\) metres. Per the official answer key, the marked answer for this question is \(\dfrac{80}{\sqrt{3}}\); the value is presented as marked in the source paper.
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