Question

The tops of two poles are of height 20 m and 14 m are connected by a wire. If the wire makes an angle 30° with the horizontal, then the length of the wire is

• A. 11 m
• B. 12 m
• C. 13 m
• D. 10 m

Answer 1

The Correct Option is B

To solve the problem, we need to determine the length of the wire connecting the tops of two poles, given their heights and the angle the wire makes with the horizontal. Let us analyze this step by step.

1. Understanding the Problem:
The tops of two poles are at heights of \( 20 \, \text{m} \) and \( 14 \, \text{m} \). The wire connecting these tops makes an angle of \( 30^\circ \) with the horizontal. We need to find the length of the wire.

2. Key Observations:
The difference in height between the two poles is:

$$ \text{Difference in height} = 20 - 14 = 6 \, \text{m} $$

This vertical difference forms one side of a right triangle, where the wire acts as the hypotenuse, and the horizontal distance between the bases of the poles forms the other side.

3. Using Trigonometry:
In the right triangle formed, the vertical difference (6 m) is the opposite side to the angle \( 30^\circ \), and the wire is the hypotenuse. Using the sine function, we have:

$$ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} $$

Since \( \sin(30^\circ) = \frac{1}{2} \), we can write:

$$ \frac{1}{2} = \frac{6}{\text{length of the wire}} $$

Let the length of the wire be \( L \). Then:

$$ \frac{1}{2} = \frac{6}{L} $$

Solving for \( L \):

$$ L = 6 \times 2 = 12 \, \text{m} $$

4. Conclusion:
The length of the wire is \( 12 \, \text{m} \).

Final Answer:
The correct option is \( {12 \, \text{m}} \).