Two poles standing on a horizontal ground are of height x meters and 40 meters respectively. The line joining their tops makes an angle of `30^(@)` with the ground and the distance between the foot of the poles is `30sqrt3` meters, then the value of x can be

Options

A. 20
B. 30
C. 10
D. 50

Answer

To solve the problem, we need to find the height \( x \) of the first pole given the height of the second pole and the distance between the poles. We will use trigonometric relationships in right triangles formed by the poles and the line joining their tops.

### Step-by-Step Solution:

1. **Identify the given information**:
- Height of the first pole = \( x \) meters
- Height of the second pole = \( 40 \) meters
- Distance between the feet of the poles = \( 30\sqrt{3} \) meters
- Angle made by the line joining the tops of the poles with the ground = \( 30^\circ \)

2. **Draw a diagram**:
- Let \( A \) be the foot of the first pole (height \( x \)).
- Let \( B \) be the foot of the second pole (height \( 40 \)).
- Let \( C \) be the top of the first pole (point \( (0, x) \)).
- Let \( D \) be the top of the second pole (point \( (30\sqrt{3}, 40) \)).
- The distance \( AB = 30\sqrt{3} \).

3. **Set up the triangles**:
- In triangle \( ACD \) (formed by the tops of the poles and the ground), we can use the tangent function:
\[ \tan(30^\circ) = \frac{CD}{AD} \]
- Here, \( CD = 40 - x \) (the vertical distance between the tops of the poles) and \( AD = 30\sqrt{3} \).

4. **Calculate using the tangent function**:
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{40 - x}{30\sqrt{3}} = \frac{1}{\sqrt{3}} \]
- Cross-multiplying gives:
\[ 40 - x = 30 \]

5. **Solve for \( x \)**:
- Rearranging the equation:
\[ x = 40 - 30 \]
\[ x = 10 \]

### Final Answer:
The value of \( x \) is \( 10 \) meters.

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