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Question

State true(1) or false(0):The angle of elevation of the top of a tower is $$30^{0}$$. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

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Solution

The given angle of elevation $$={ 30 }^{ o }.$$ 
Let the height of the tower $$=h$$ and the viewer be at a distance of $$x$$ from the foot of the tower.
Then, $$\cfrac { h }{ x } =\tan{ 30 }^{ o }=\cfrac { 1 }{ \sqrt { 3 }  }$$ ........(i)
If the height of the tower is doubled then the new height $$=2h.$$ 
Let the angle of elevation of the top be $$\theta.$$ 
Then, $$\tan\theta =\cfrac { 2h }{ x } =2\times \cfrac { 1 }{ \sqrt { 3 }  } =\cfrac { 2 }{ \sqrt { 3 }  }$$ ......(from i)...........(ii)
But if the angle of elevation doubles then it should be $$=\theta ={ 2\times 30 }^{ o }={ 60 }^{ o }.$$ 
Then, $$\tan\theta =\tan{ 60 }^{ o }=\sqrt { 3 }$$ ........(iii).
Comparing (ii) & (iii), there is a contradiction.
$$ \therefore$$ The assertion is incorrect.

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Mathematics

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