The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant, the angles of elevation of a balloon from these windows are observed to be 60 ° and 30 ° , respectively, find the height of the balloon above the ground.

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The lower window of a house is at a height of $2 m$ above the ground and its upper window is $4 m$ vertically above the lower window. At certain instant, the angles of elevation of a balloon from these windows are observed to be $60 °$ and $30 °$

, respectively, find the height of the balloon above the ground.

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4 m

6 m

8 m

2 m

answer is D.

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Detailed Solution

Given, height of lower window, h1=2m.
Height of upper window, h2=4m.
Angle of elevation of balloon from the lower window,

θ_{1} = 60 °

And angle of elevation of balloon from the upper window,

θ_{2} = 30 °

Consider the figure,
Question Image

Considering

∆ ADE

tan θ = p e r p e n d i c u l a r b a s e tan 30 ° = H x ⇒ x = H tan 30 ° ⇒ x = H 3 ∵ tan 30 ° = 13

Similarly, consider,

∆ AFG,

tan θ = p e r p e n d i c u l a r b a s e ⇒ tan 60 ° = H + 4 x ⇒ x = H + 4 tan 60 ° ⇒ x = H + 4 3 ∵ tan 60 ° = 3

Comparing the value of x,

H 3 = H + 4 3 ⇒ 3 H = H + 4 ⇒ 2 H = 4 ⇒ H = 2

Therefore, the height above the upper window is 2m.
So, the total height from the ground to the balloon is 2+4+6=8m
Therefore, option 4 is correct.

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