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From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be $30^{o}$ and $45^{o}$ . Find the height of the hill.

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1.365

1.5

1.7

1.1

answer is A.

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Detailed Solution

Given that,
Angles of depression =

30^{o}

and

45^{o}

We have the formula,

tanθ = \frac{Height}{Distance}

We have to find the height of the hill,
Considering the distance between the nearer kilometer stone and the hill be '

x

' km.
Then the distance between the farther kilometer stone and the hill will be '

1 + x

' km as both are on the same side of the hill.
In triangle APB,

\tan 45^{0} = \frac{h}{x}

\Rightarrow

1 = \frac{h}{x}

\Rightarrow

h = x

…..(1)
In triangle AQB,

\tan 30^{0} = \frac{h}{1 + x}

\Rightarrow

\frac{1}{\sqrt{3}} = \frac{h}{1 + x}

\Rightarrow

1 + x = \sqrt{3} h

From equation (1),

1 + h = \sqrt{3} h

\Rightarrow 1 = \sqrt{3} h - h

\Rightarrow

h = \frac{1}{\sqrt{3} - 1}

\Rightarrow

h = 1.365 km

(

\sqrt{3} = 1.732)

Thus, the height is 1.365km.
Therefore the correct option is 1.

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