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Question
A statue, 1.6 m tall, stands on a top of pedestal, from a point on the ground, the angle of elevation of the top of statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
Solution
Let AB be the statue, BC be the pedestal, and D be the point on the ground from where the elevation angles are to be measured.
In ΔBCD,
`("BC")/("CD")` = tan 45°
`("BC")/("CB")` = 1
BC = CD
In ΔACD,
`("AB"+"BC")/("CD")` = tan 60°
`("AB"+"BD")/("BC") = sqrt3`
`1.6+"BC" = "BC"sqrt3`
`"BC"(sqrt3-1) = 1.6`
BC = `((1.6)(sqrt(3)+1))/((sqrt(3)-1)(sqrt(3)+1))`
= `(1.6(sqrt(3)+1))/((sqrt(3))^2-(1)^2)`
= `(1.6(sqrt3+1))/2`
= `0.8(sqrt3+1)`
Therefore, the height of the pedestal is 0.8 `(sqrt3+1)` m.
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