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Question
The angle of elevation of a cloud from a point h metres above a lake is β. The angle of depression of its reflection in the lake is 45°. The height of location of the cloud from the lake is
Options
`("h"(1 + tan beta))/(1 - tan beta)`
`("h"(1 - tan beta))/(1 + tan beta)`
h tan(45° − β)
none of these
Solution
`("h"(1 + tan beta))/(1 - tan beta)`
Explanation;
Hint:
Consider the height of the cloud PC b x
PD = x − h
Let BC be y
In the right ΔADP, tan β = `"PD"/"AD"`
tan β = `(x - "h")/y`
⇒ y = `(x - "h")/(tan beta)` ...(1)
In the right ΔAQD, tan 45° = `"DQ"/"AD"`
1 = `(x + "h")/y`
⇒ y = x + h ...(2)
From (1) and (2) we get,
`(x - "h")/(tan beta)` = x + h
⇒ (x + h) tan β = x − h
⇒ x tan β + h tan β = x − h
h + h tan β = x − x tan β
⇒ h(1 + tan β) = x(1 − tan β)
x = `("h"(1 + tan beta))/((1 - tan beta))`
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