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The top of a $15$ meters high tower makes an angle of elevation of $60 °$ with the bottom of an electric pole and angle of elevation of ${30 °}^{}$ with the top of the pole. What is the height of the electric pole?

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5

meters

8

meters

10

meters

12

meters

answer is C.

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Detailed Solution

It is given that the tower makes an angle of elevation of

60 °

with the bottom of an electric pole. Also it makes an angle of elevation of 30

°

with the top of the pole.
According to the given data the figure has been drawn below.
Question Image

Let

AB

be the tower and

CD

be the electric pole.
Then

∠ ACB = 60 °, ∠ EDB = 30 °

and

AB = 15

m.
Let,

CD = h .

Then

BE

\Rightarrow (AB - AE)

\Rightarrow (AB - CD)

\Rightarrow (15 - h)

…..(1)
We have,
In

Δ A B C,

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

\frac{AB}{AC} = \tan 60 °

\frac{AB}{AC}

= \sqrt{3}

(\tan 60 ° = \sqrt{3})

\Rightarrow AC = \frac{AB}{\sqrt{3}}

\Rightarrow AC = \frac{15}{\sqrt{3}}

Δ B D E,

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

\frac{BE}{DE} = \tan 30 °

\frac{BE}{DE} = \frac{1}{\sqrt{3}}

(\tan 30 ° = \frac{1}{\sqrt{3}})

\Rightarrow DE = BE \times \sqrt{3}

\Rightarrow DE = \sqrt{3} (15 - h)

\Rightarrow \frac{15}{\sqrt{3}} = \sqrt{3} (15 - h)

(DE=AC since they are the opposite sides of the rectangle AEDC)

\Rightarrow 15 = 3 (15 - h)

\Rightarrow 15 = 45 - 3 h

\Rightarrow 3 h = 45 - 15

\Rightarrow 3 h = 30

\Rightarrow h = \frac{30}{3}

\Rightarrow h = 10

So, the height of the pole is 10 m.
So, the correct option is 3.

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The top of a 15 meters high tower makes an angle of elevation of 60° with the bottom of an electric pole and angle of elevation of 30°with the top of the pole. What is the height of the electric pole?

Detailed Solution

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The top of a $15$ meters high tower makes an angle of elevation of $60 °$ with the bottom of an electric pole and angle of elevation of ${30 °}^{}$ with the top of the pole. What is the height of the electric pole?