An observer is positioned 200 m above a lake's surface. From this position, the angle of elevation to a cloud is 30° and the angle of depression to the reflection of the cloud in the lake is 60°. What is the height of the cloud above the lake's surface?

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To determine the cloud's height, we'll use trigonometric principles with angles of elevation and depression.

Let's define **h** as the height of the cloud above the lake and **d** as the horizontal distance from the observation point to the point directly below the cloud.

For the **angle of elevation** (30°):
The vertical distance from the observer to the cloud is (h - 200) meters.
tan 30° = (h - 200)/d
1/√3 = (h - 200)/d
d = √3(h - 200)

For the **angle of depression** (60°):
The vertical distance from the observer to the reflection is (200 + h) meters.
tan 60° = (200 + h)/d
√3 = (200 + h)/d
d = (200 + h)/√3

Since both expressions for **d** must be equal:
√3(h - 200) = (200 + h)/√3

Multiplying both sides by √3:
3(h - 200) = 200 + h

Expanding and solving:
3h - 600 = 200 + h
3h - h = 200 + 600
2h = 800
h = 400

Therefore, the height of the cloud above the lake is **400 meters**

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