The length of the shadow of a tower standing on level ground four to be `2x` metres longer when the sun's elevation is `30^(@)` than where it was `45^(@)`. The height of the tower is

Options

A. `(2sqrt(3)x)m`
B. `(3sqrt(2)x)m`
C. `(sqrt(3) 1) x m`
D. `(sqrt(3) + 1) x m`

Answer

D

Explanation

Let h m be the height of the tower and let a m and`(a+2x)`m be the lengths of the shadows of the tower when sun's elevation is `45^(@)` and `30^(@)` respectively. Then,
`h/a = tan 45^(@) = 1 rArr a = h`.
`h/(a+2x) = tan 30^(@) = 1/(sqrt(3)) rArr h/(h+2x) = 1/(sqrt(3))` .
`:. sqrt(3) h = h + 2x rArr 2x = (sqrt(3) - 1) h rArr h = (2x)/((sqrt(3) - 1))`
`:. h = {(2x)/((sqrt(3) - 1)) xx ((sqrt(3) + 1))/((sqrt(3) + 1))} = x (sqrt(3) + 1)`.