Final answer:
To find the cloud's height above a lake, a right triangle is formed using the angle of elevation and angle of depression. The cloud's height is the sum of the observer's height and the calculated distance using trigonometry, resulting in approximately 163.92 meters.
Step-by-step explanation:
The problem given involves finding the height of a cloud using the angle of elevation and the angle of depression of its reflection in a lake. It applies principles of trigonometry relating to right triangles formed between the observer, the cloud, and its reflection.
An observer is 60 meters above a lake surface. The angle of elevation to the cloud is 30°. The angle of depression to the reflection is 60°, forming a right triangle between the observer, the cloud, and its reflection.
The triangle formed by the observer, the reflection, and a point directly underneath the cloud (where the perpendicular line from the cloud meets the lake) is a 30-60-90 right triangle. We know that the ratios of the sides are in the proportion 1:√3:2 for the sides opposite the 30°, 60°, and 90° angles respectively. The distance from the observer to the point under the cloud is 60 m, which corresponds to the side opposite the 30° angle. Therefore, the distance from the cloud to its reflection (the side opposite the 60° angle) is 60√3 meters.
The total height of the cloud above the lake is then the observer's height plus this distance, which is 60 m + 60√3 m. Simplifying that, we get approximately 60 + 103.92 = 163.92 m as the height of the cloud above the lake.