The tops of two poles of height 24 m and 36 m are connected by a wire. If the wire makes an angle of `60^@` with the horizontal, then the length of the wire is

Options

A. 8V3 m
B. 8 m
C. 6V 3m
D. 6 m

Answer

To find the length of the wire connecting the tops of two poles of heights 24 m and 36 m, which makes an angle of 60 degrees with the horizontal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Heights of the Poles:**
- Let the height of pole A (36 m) be represented as \( h_1 = 36 \, \text{m} \).
- Let the height of pole B (24 m) be represented as \( h_2 = 24 \, \text{m} \).

2. **Calculate the Height Difference:**
- The vertical distance between the tops of the two poles is given by:
\[ h = h_1 - h_2 = 36 \, \text{m} - 24 \, \text{m} = 12 \, \text{m} \]

3. **Draw a Right Triangle:**
- The wire forms the hypotenuse of a right triangle where:
- The vertical side (opposite side) is the height difference \( h = 12 \, \text{m} \).
- The angle with the horizontal is \( 60^\circ \).

4. **Use the Sine Function:**
- In a right triangle, the sine of an angle is defined as the ratio of the opposite side to the hypotenuse. Therefore, we can write:
\[ \sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L} \]
- Where \( L \) is the length of the wire (hypotenuse).

5. **Substitute Known Values:**
- We know \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \) and \( h = 12 \, \text{m} \):
\[ \frac{\sqrt{3}}{2} = \frac{12}{L} \]

6. **Rearrange to Solve for \( L \):**
- Cross-multiplying gives:
\[ L \cdot \frac{\sqrt{3}}{2} = 12 \]
- Therefore:
\[ L = \frac{12 \cdot 2}{\sqrt{3}} = \frac{24}{\sqrt{3}} \]

7. **Rationalize the Denominator:**
- To express \( L \) in a more standard form, multiply the numerator and denominator by \( \sqrt{3} \):
\[ L = \frac{24 \sqrt{3}}{3} = 8\sqrt{3} \, \text{m} \]

### Final Answer:
The length of the wire is \( 8\sqrt{3} \, \text{m} \).