Question

The angle of elevation of a cloud from a height h above the level of water in a lake is α and the angle of depression of its image in the lake is β. Find the height of the cloud above the surface of the lake :

A. h sin(β−α)sin(α+β)
B. h sin α
C. h sin(α+β)/sin(β−α)
D. None of these

Answer 1

The height of the cloud above the surface of the lake is calculated using angles of elevation and depression, trigonometry, and algebra, which results in option C: h sin(α+β)/sin(β-α).

Step-by-step explanation:

To find the height of the cloud above the surface of the lake, we can use the concept of angles of elevation and depression in trigonometry. Since the angle of elevation from the observer to the cloud is α and the angle of depression to its reflection in the water is β, we can consider two right-angled triangles sharing a common side, which is the height of the cloud above the height h.

Let x be the height of the cloud above the height h. By using trigonometry, we can express x in terms of the given angles and h. The total height of the cloud above the lake is h + x.

For the angle of elevation (α):

tan(α) = x / d

For the angle of depression (β):

tan(β) = (h + x) / d

By solving these two equations together, we can isolate x and express the total height of the cloud above the water surface. After a series of algebraic manipulations, the result comes out to be:

h sin(α+β)/sin(β-α) (Option C).