From the top of a 50 m high tower, the angle of depression to a car on the ground is 30° and the angle of elevation to a balloon directly above the car is 60°. Find the height of the balloon from the ground.
200 metres
Let the horizontal distance from the foot of the tower to the car be d. The tower top is 50 m high.
Angle of depression to the car is 30°, so from the top \(\tan 30^{\circ}=\dfrac{50}{d}\).
Thus \(d=\dfrac{50}{\tan 30^{\circ}}=50\sqrt{3}\) m.
The balloon is directly above the car. From the tower top its angle of elevation is 60°, so \(\tan 60^{\circ}=\dfrac{h-50}{d}\), where h is the balloon height.
So \(h-50=d\tan 60^{\circ}=50\sqrt{3}\times\sqrt{3}=150\).
Therefore \(h=150+50=200\) m.
Hence, the height of the balloon from the ground is 200 metres.
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