Question

From a ballon vertically above a straight road,the angle of depression of two cars on the same side at an instant are found to be 45 and 60 degrees.If the cars are 100m apart . Find the height of the balloon

Answer 1

Let the two angle be x = 45° and y = 60° respectively.
And the distance between the two cars is d = 100 m
Height of balloon is AB  = h meter (See the figure)
Let BC = x meter
In triangle ABO,
tan 45° = AB/BO  ⇒ 1 = h/(x+100) ⇒ h = x + 100 ........(1)
In triangle ABC,tan 60° = AB/BC ⇒ √3 = h/x⇒ h = x√3 ...... (2)
As h = x + 100
⇒ x + 100 = x√3
⇒ 100 = √3 x - x
⇒ 100 = x(√3 - 1)
⇒ x = 100/(√3 - 1)
⇒ x = 100 (√3 + 1)/(√3 - 1) (√3 + 1)/(√3 + 1)   
{Multiplied by √3 +1 with both numerator and denominator}
⇒ x = 100(1.73 + 1)/2 
⇒ x = 100(2.73/2)
⇒ x = 273/2
⇒ x = 136.5 m 
So, BC = 136.5 metre
So, h = x + 100
h = 136.5 + 100h = 236.5 metre

:) Hope this Helps!!!