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From a balloon vertically above a straight road, the angle of depression of two cars at an instant are found to be $45 °$ and $60 ° .$ If the cars are $100 m$ apart, then the height of the balloon is

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246.6 m.

256.6 m.

226.6 m.

236.6 m.

answer is B.

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Detailed Solution

We are given that a balloon is vertically above a straight road, from where the angle of depression of two cars at an instant are found to be

45 °

and

60 ° .

Consider the figure,
Question Image

We know,

tanθ = \frac{Perpendicular}{Base}

\tan 60 ° = \sqrt{3}

\tan 30 ° = \frac{1}{\sqrt{3}}

Let the distance BQ is equal to x.
Therefore in triangle PBQ,

\tan 60 ° = \frac{PQ}{BQ}

\Rightarrow \tan 60 ° = \frac{h}{x}

\Rightarrow h = xtan 60^{0}

………..(1)
Similarly PAQ,

\tan 45 ° = \frac{PQ}{AQ}

\Rightarrow \tan 45 ° = \frac{h}{(x + 100)}

\Rightarrow h = (x + 100) \tan 45^{0}

As both heights are equal so determine the value of x,

x tan 60 ° = (x + 100) tan 45 ° ⇒ 3 x = x + 100 ⇒ x = 100 3 − 1

Now, substituting the value of x in equation 1,

h = 100 3 − 1 tan 60 ° ⇒ h = 100 3 − 1 3 ⇒ h = 236.6

The height of the balloon is 236.6 m.
Therefore, option 1 is correct.

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