From the observation point P, the angle of elevation to the cloud C is α. The vertical distance from P to C is H-h. Thus, tan α = (H-h)/(x). The angle of depression to the reflection C' is β. The reflection C' is at a depth H below the lake surface, so its distance from the observation point's horizontal line is H+h. Thus, tan β = (H+h)/(x).

From the observation point P, the angle of elevation to the cloud C is α. The vertical distance from P to C is H-h. Thus, tan α = (H-h)/(x). The angle of depression to the reflection C' is β. The reflection C' is at a depth H below the lake surface, so its distance from the observation point's horizontal line is H+h. Thus, tan β = (H+h)/(x).

By equating the two expressions for x, we form a single equation involving H, h, tan α, and tan β. We then expand and rearrange the terms to isolate H on one side, leading to the desired formula for the height of the cloud.

By equating the two expressions for x, we form a single equation involving H, h, tan α, and tan β. We then expand and rearrange the terms to isolate H on one side, leading to the desired formula for the height of the cloud.

The angle of elevation of a cloud from a point $h$ meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$ . Prove that the height of the cloud is: $\frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha}$

Answer: The height of the cloud is $\frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha}$

Step-by-step solution

Step 1: Define variables and set up the diagram

We begin by defining the key points and distances involved in the problem. Let $C$ represent the cloud, $C'$ its reflection in the lake, and $P$ the observation point. The lake surface is denoted by $L$ . The height of the observation point $P$ above the lake surface is given as $h$ . We assume the height of the cloud above the lake surface is $H$ and the horizontal distance from the observation point to the vertical projection of the cloud on the lake surface is $x$ .

Step 2: Formulate equations using angles of elevation and depression

From the observation point $P$ , the angle of elevation to the cloud $C$ is $\alpha$ . The vertical distance from $P$ to $C$ is $H-h$ . Thus, $\tan \alpha = \frac{H-h}{x}$ . The angle of depression to the reflection $C'$ is $\beta$ . The reflection $C'$ is at a depth $H$ below the lake surface, so its distance from the observation point's horizontal line is $H+h$ . Thus, $\tan \beta = \frac{H+h}{x}$ .

Step 3: Solve for x from both equations

We can rearrange both equations (1) and (2) to solve for $x$ . This will allow us to equate the expressions for $x$ in the next step.

Step 4: Equate the expressions for x and solve for H

By equating the two expressions for $x$ , we form a single equation involving $H$ , $h$ , $\tan \alpha$ , and $\tan \beta$ . We then expand and rearrange the terms to isolate $H$ on one side, leading to the desired formula for the height of the cloud.

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