from a balloon vertically above a straight road the ankle of depression of two cars at an instant are found to 45 degree and 60 degree.if the cars are 100m apart find the height of the balloon

It is given that distance between the 2nd car and 1st car is 100 m

let the distance between the baloon and ground be AB

and

let the distance between the bottom of the baloon and nearer car be x

tan60⁰ = AB / x

√3 =  AB / x

AB = x√3               ______________(i)

tan45⁰ = AB / x+100 

1 = AB / x+100

AB = x+100               _______________(ii)

on comparing (i)&(ii)

we get ,

x√3 = x+100

x√3 - x = 100

x (√3-1) = 100

x = 100 / (√3-1)

x = 100(√3+1) / (√3-1)(√3+1)

x = 100( √3+1) / 3-1

x = 100(√3+1) / 2

x = 50(√3+1)

now,

AB = x√3                              [from (i)] 

AB = 50(√3+1)√3

AB = 50(√3+3)

hence, the height of the balloon is 50(√3+3)