An earthing wire connected to the top of an electricity pole has its other end inside the ground. The foot of the wire is 1.5 m away from the pole and the wire is making an angle of `60^@` with the level of the ground. Determine the length of wire.

Options

A. 2 m
B. 3 m
C. `sqrt3`m
D. `sqrt(3//2)`m

Answer

To determine the length of the earthing wire connected to the top of an electricity pole, we can use trigonometric principles. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
We have a right triangle formed by:
- The pole (vertical side)
- The distance from the pole to the foot of the wire (horizontal side)
- The wire itself (hypotenuse)

Given:
- The distance from the pole to the foot of the wire (horizontal side) = 1.5 m
- The angle the wire makes with the ground = 60 degrees

### Step 2: Identify the Sides of the Triangle
Let:
- A = Top of the pole
- B = Foot of the wire on the ground
- C = Point where the wire is connected to the ground

In triangle ABC:
- AB is the height of the pole (unknown)
- BC = 1.5 m (horizontal distance)
- AC = Length of the wire (hypotenuse, unknown)

### Step 3: Use Trigonometric Ratios
Since we know the angle and one side, we can use the cosine function:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Here, the adjacent side is BC (1.5 m), and the hypotenuse is AC (length of the wire).

Using the angle:
\[ \cos(60^\circ) = \frac{1.5}{AC} \]

### Step 4: Substitute the Value of Cosine
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
Now, substituting this into the equation:
\[ \frac{1}{2} = \frac{1.5}{AC} \]

### Step 5: Solve for AC
Cross-multiplying gives:
\[ AC \cdot \frac{1}{2} = 1.5 \]
\[ AC = 1.5 \cdot 2 \]
\[ AC = 3 \text{ m} \]

### Conclusion
The length of the wire (AC) is **3 meters**.

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