Two poles of equal height are standing opposite to each other on either side of a road which is 100 m wide .From a point between them on road , angle of elevation of their tops are `30^(@)and60^(@)` .The height of each pole (in metre ) is

Options

A. `25sqrt(3)`
B. `20sqrt(3)`
C. `28sqrt(3)`
D. `30sqrt(3)`

Answer

To solve the problem step by step, we will use trigonometric principles and the information provided in the question.

### Step 1: Understand the Setup
We have two poles of equal height (let's denote the height as \( h \)) standing on either side of a road that is 100 meters wide. We denote the points as follows:
- Let \( A \) and \( B \) be the tops of the poles.
- Let \( C \) be the point on the road where the angles of elevation are measured.
- The distance from point \( C \) to the base of pole \( A \) is \( x \) meters, and the distance from point \( C \) to the base of pole \( B \) is \( 100 - x \) meters.

### Step 2: Set Up the Trigonometric Relationships
From the point \( C \):
1. The angle of elevation to the top of pole \( A \) is \( 30^\circ \).
2. The angle of elevation to the top of pole \( B \) is \( 60^\circ \).

Using the tangent function, we can write:
- For pole \( A \):
\[ \tan(30^\circ) = \frac{h}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \tag{1} \]

- For pole \( B \):
\[ \tan(60^\circ) = \frac{h}{100 - x} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{h}{100 - x} \implies h = \sqrt{3}(100 - x) \tag{2} \]

### Step 3: Equate the Two Expressions for \( h \)
From equations (1) and (2), we can set them equal to each other:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(100 - x) \]

### Step 4: Solve for \( x \)
Cross-multiplying gives:
\[ x = 3(100 - x) \]
\[ x = 300 - 3x \]
\[ x + 3x = 300 \]
\[ 4x = 300 \implies x = 75 \text{ meters} \]

### Step 5: Substitute \( x \) Back to Find \( h \)
Now substitute \( x = 75 \) back into either equation (1) or (2) to find \( h \). Using equation (1):
\[ h = \frac{75}{\sqrt{3}} = 25\sqrt{3} \text{ meters} \]

### Conclusion
The height of each pole is \( 25\sqrt{3} \) meters.

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