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State the true or false.

The length of the shadow of a tower standing on the level plane is found to be $2 x$ metres longer when the sun’s altitude is 30 than when it was 45, then the height of tower $x (\sqrt{3} + 1),$

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answer is A.

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Detailed Solution

Given,
Suppose that
Height of tower = AB
The two points on ground = C and D

CD = 2 x m

∠ ADB = 45^{\circ}

∠ ACB = 30^{\circ}

First, we have drawn the figure as per condition
Question Image

In the triangle

△ ABD

\frac{AB}{BD} = \tan 45^{\circ}

\frac{AB}{BD} = 1

\Rightarrow h = BD

Now, in the triangle

△ {ABC}_{r}

\frac{AB}{BC} = \tan 30^{\circ}

\Rightarrow \frac{h}{BC} = \frac{1}{\sqrt{3}}

\Rightarrow BC = \sqrt{3} h

\Rightarrow CD + h = \sqrt{3} h

\Rightarrow 2 x = (\sqrt{3} - 1) h

Therefore,

h = \frac{2 x}{\sqrt{3} - 1}

h = \frac{2 x (\sqrt{3} + 1)}{(\sqrt{3})^{2} - 1}

h = x (\sqrt{3} + 1) m

Hence, the height of the tower

= x (\sqrt{3} + 1) m

this statement is true.
Correct option is 1.

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