From a point P on the ground the angle of elevation of the top of a tower is `30^(@)` and that of the top a flagstaff fixed on the top of the tower is `60^@`. If the length of the flagstaff is 5 m, find the height of the tower.

Options

A. `1.5` m
B. `3.5` m
C. `2.5` m
D. `4.5`m

Answer

C

Explanation

Let AB be the tower and `BC` be the flagstaff.
Let O be a point on the ground such that `angleAOB = 30^(@)` and `angleAOC = 60^(@), BC = 5 m` (given).
Let `AB = h` metres and `OA = x` metres .
From right `DeltaOAB`, we have
`(OA)/(AB) = cot 30^(@) = sqrt(3) rArr x/h = sqrt(3)`.
`rArr = hsqrt(3)"........"(i)`.
From right `DeltaOAC`, we have
`(OA)/(AC) = cot 60^(@) = 1/(sqrt(3))`
`rArr(x)/(h+5) = 1/(sqrt(3)) rArr x = (h+5)/(sqrt(3))"........."(ii)`
Equating the values of x from (i) and (ii), we get
`hsqrt(3) = (h+5)/(sqrt(3)) rArr 3h = h+5 rArr 2h = 5 rArr h = 2.5`.
Hence the height of the tower is `2.5` metres.