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The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of $100 m$ from its base is ${45 °}^{} .$ If the angle of elevation of the top of the complete pillar at the same point is to be $60^{o},$ then the height of the incomplete pillar is to be increased by

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50 \sqrt{2} m

100 m

100 (\sqrt{3 - 1}) m

100 (\sqrt{3 + 1}) m

answer is C.

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Detailed Solution

It is given the angle of elevation of the top is

45^{o}

and horizontal distance is 100m.
According to the given data the figure has been drawn below.
Question Image

Let, BC be the incomplete and BD be the complete pillar .
In triangles △ ABC and △ABD we have been given

BC = 100

m , and

∠ CAB = {45 °}^{}

∠ DAB = {60 °}^{}

.
Let

CD = x

m.
In

ΔABC

tan θ = P e r p e n d i c u l a r b a s e

\tan {45 °}^{} = \frac{BC}{AB}

\Rightarrow 1 = \frac{BC}{100}

(\tan 45 ° = 1)

\Rightarrow BC = 100 m

In △

ABD,

tan θ = P e r p e n d i c u l a r b a s e

\tan {60 °}^{} = \frac{BD}{AB}

(\tan 60 ° = \sqrt{3})

\Rightarrow \sqrt{3} = \frac{BD}{100}

\Rightarrow BD = 100 \sqrt{3}

m…(1)
Then,

BD = BC + CD

= 100 + x

Now BD =

100 \sqrt{3}

( from equation 1)

\Rightarrow 100 \sqrt{3} = 100 + x

\Rightarrow x = 100 (\sqrt{3} - 1)

m
Hence, height of the incomplete pillar is to be increased by

100 (\sqrt{3} - 1)

m.
So, the correct option is 3.

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