A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at `B` is twice and at `C` is thrice that at `A` . If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then find the height of balloon above the road.

Explanation

Let P be the balloon at height h above the road .

From the above figure ,
`x=hcot3alpha ...(1)
(x+100)=hcot2alpha ...(2)
(x+300)=hcotalpha …(3)`
Subtracting (1) from (2) ,we get
100=h(cot2alpha-cot3alpha) ,
`therefore 100 =h(sinalpha/(sin3alpha sin2alpha))` ...(4) subtracting (2)from(3) , we get
` 200 =h (cotalpha-cot2alpha)`
`therefore 200 =h (sinalpha /(sin2alpha sinalpha ))` ...(5)
Dividing (5) by (4), we get
`(sin3alpha)/(sinalpha) =200/100=2`
rArr`3sinalpha-4sin^3alpha -2sinalpha =0`
`rArr sin^2alpha =1/4 =sin^2""pi/6`
` rArr alpha = pi/6`
using the value in (5) , we get
`therefore h=200sin""(pi)/3=200sqrt3/2=100sqrt3`