The angle of elevation of a cloud from a point h meters above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the height of the cloud is: tanβ−tanαh(tanβ+tanα)
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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 α. The vertical distance from P to C is H-h. Thus, tanα=xH−h. 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, tanβ=xH+h.
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, tanα, and tanβ. We then expand and rearrange the terms to isolate H on one side, leading to the desired formula for the height of the cloud.