Question

There are two windows in a house. A window of the house is at a height of 1.5 m above the ground and the other window is 3 m vertically above the lower window. Ram and Shyam are sitting inside the two windows. At an instant, the angle of elevation of a balloon from these windows is observed as 45° and 30° respectively. Find the height of the balloon from the ground.

Options

• 7.598 m

• 8.269 m

• 7.269 m

• 8.598 m

Answer 1

8.598 m

Explanation:

 

Then

AP = 1.5 m

And

AB = 3 m

AP = DQ = 1.5 m and BA = ED = 3 m

Let the height of balloon from ground be h,

Then CE = (h − 4.5)m

In right triangle ADC

`"tan"  45^circ = "CD"/"AD"`

`=> 1 = (("h" - 4.5) + 3)/"AD"`

`=> 1 = ("h" - 1.5)/"AD"`

`=> "AD" = "h" - 1.5`

In right triangle CEB

`"tan"  30^circ = "CE"/"BE"`

`=> 1/sqrt3 = ("h" - 4.5)/("h" - 1.5)`

`=> 1/sqrt3 = ("h" - 1.5)/("h" - 4.5)`

`=> "h" - 1.5 = sqrt3 ("h" - 4.5)`

`=> "h" - 1.5 = sqrt3 "h" - 4.5 sqrt3`

`=> 4.5 sqrt3 - 1.5 = sqrt3 "h" - "h"`

`"h" = (4.5 sqrt3 - 1.5)/(sqrt3 - 1)`

`"h" = (1.5 (3 sqrt3 - 1))/(sqrt3 - 1)`

`=> "h" = (1.5 (3sqrt3 - 1))/(sqrt3 - 1). ((sqrt3 + 1))/(sqrt3 + 1)`

`=> "h" = (1.5 (9 - sqrt3 + 3 sqrt3 - 1))/2`

`=> "h" = (1.5 (8 + 2 sqrt 3))/2`

`=> "h" = 1.5 (4 + sqrt3)`

`=> "h" = 8.598  "m"`