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Question
From the top of a lighthouse, the angle of depression of two ships on the opposite sides of it is observed to be 30° and 60°. If the height of the lighthouse is h meters and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ships is `(4"h")/sqrt(3)` m
Solution
A and C be the position of two ships.
Let AB be x and BC be y.
Distance between the two ships is x + y
In the right ∆ABD, tan 60° = `"BD"/"AB"`
`sqrt(3) = "h"/x`
x = `"h"/sqrt(3)` ...(1)
In the right ∆BCD,
tan 30° = `"BD"/"BC"`
`1/sqrt(3) = "h"/y`
y = `sqrt(3)`h
Distance between the two ships (x + y) = `"h"/sqrt(3) + sqrt(3)"h"`
= `("h" + 3"h")/sqrt(3)`
= `(4"h")/sqrt(3)`
Hence it is verified.
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