The top 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. 6 m
B. `8sqrt(3)m`
C. 8 m
D. `6sqrt(3)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 1: Understand the Setup
We have two poles:
- Pole 1 (height = 24 m)
- Pole 2 (height = 36 m)

The difference in height between the two poles is:
\[ \text{Height difference} = 36 \, \text{m} - 24 \, \text{m} = 12 \, \text{m} \]

### Step 2: Draw a Diagram
Draw a right triangle where:
- The vertical side (perpendicular) represents the height difference (12 m).
- The horizontal side (base) is the distance between the two poles.
- The hypotenuse represents the length of the wire.

### Step 3: Use Trigonometric Ratios
Since the wire makes an angle of 60 degrees with the horizontal, we can use the sine function:
\[ \sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, the opposite side is the height difference (12 m), and the hypotenuse is the length of the wire (let's denote it as \( L \)):
\[ \sin(60^\circ) = \frac{12}{L} \]

### Step 4: Substitute the Value of Sine
We know that:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Substituting this into the equation gives:
\[ \frac{\sqrt{3}}{2} = \frac{12}{L} \]

### Step 5: Solve for \( L \)
Cross-multiply to solve for \( L \):
\[ \sqrt{3} \cdot L = 2 \cdot 12 \]
\[ \sqrt{3} \cdot L = 24 \]
Now, divide both sides by \( \sqrt{3} \):
\[ L = \frac{24}{\sqrt{3}} \]

### Step 6: Rationalize the Denominator
To simplify \( L \), multiply the numerator and denominator by \( \sqrt{3} \):
\[ L = \frac{24 \cdot \sqrt{3}}{3} = 8\sqrt{3} \]

### Final Answer
The length of the wire is:
\[ L = 8\sqrt{3} \, \text{m} \]

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