The angle of elevation of the top of a tower from the top and bottom of a building of height 'a' are `30^@` and `45^@` respectively. If the tower and the building stand at the same level , the height of the tower is

Options

A. `(a(3+sqrt3))/2`
B. `a(sqrt3+1)`
C. `sqrt3a`
D. `a(sqrt3-1)`

Answer

A

Explanation

In `triangle's` PCB and POA, we have
`tan 30^@ = (PC)/(BC)` and `tan 45^@ =(OP)/(OA)`
`rArr 1/sqrt3-(PC)/(OA)` and `1-(OP)/(OA)`
`rArr 1/sqrt3=(PC)/(OP) rArr PC=(OP)/sqrt3`

`therefore OP=OC+PC = a + (OP)/sqrt3`
`rArr OP(1-1/sqrt3)=a rArr OP=(sqrt3a)/(sqrt3-1)=(sqrt3(sqrt3+1)a)/2`