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Question
The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30° and 45° respectively. Find the height of the multistoried building and the distance between the two buildings.
Solution
Let AD be the multistoried building of height hm. And the angle of depression of the top and bottom are 30° and 45°. We assume that BE = 8, CD = 8 and BC = x, ED = x and AC = h − 8. Here we have to find height and distance of the building.
We use trigonometric ratio.
In ΔAED,
`=> tan E = (AD)/(DE)`
`=> tan 45^@= (AD)/(DE)`
`=> 1 = h/x`
=> x = h
Again in Δ ABC
`=> tan B = (AC)/(BC)`
`=> tan 30^@ = (h - 8)/x`
`=> 1/sqrt3 = (h - 8)/x`
`=> hsqrt3 - 8sqrt3 = x`
`> hsqrt3 - 8sqrt3 = h`
`=> h(sqrt3 - 1) = 8sqrt3`
`=> h = (8sqrt3)/(sqrt3 - 1) xx (sqrt3 + 1)/(sqrt3 + 1)`
`=> h = (24 + 8sqrt3)/2`
`=> h = (4(3 + sqrt3))`
And
`=> x = 4(3 + sqrt3)`
Hence the required height is `4(3 + sqrt3)` meter and distance is `4(3 + sqrt3)` meter
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