The tops of two poles of height 22 m and 31 m are connected by a wire…

The correct answer is Option (1) → $6\sqrt{3}$

1. Identify the Vertical Difference

The two poles have heights of 31 m and 22 m. When a wire connects the tops, the vertical distance ($h$) between the two tops is the difference in their heights:

$h = 31\text{ m} - 22\text{ m} = 9\text{ m}$

2. Form a Right-Angled Triangle

Imagine a horizontal line drawn from the top of the shorter pole (22 m) to the taller pole (31 m). This forms a right-angled triangle where:

• The Perpendicular ($P$) is the height difference: 9 m.
• The Hypotenuse ($L$) is the length of the wire (which we need to find).
• The Angle of elevation ($\theta$) with the horizontal is 60°.

3. Apply Trigonometry

We use the sine function, which relates the perpendicular and the hypotenuse:

$\sin(\theta) = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$

$\sin(60^\circ) = \frac{9}{L}$

Since $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, we substitute the value:

$\frac{\sqrt{3}}{2} = \frac{9}{L}$

4. Solve for $L$

$L = \frac{9 \times 2}{\sqrt{3}}$

$L = \frac{18}{\sqrt{3}}$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$:

$L = \frac{18 \times \sqrt{3}}{3}$

$L = 6\sqrt{3}$

Final Answer:

The length of the wire is $6\sqrt{3}$ m.