A pole is standing erect on the ground which is horizontal. The tip of the poke is tied tight with a rope of length ` sqrt(12)` m to a point on the ground. If the rope is making `30^(@)` angle with the horizontal , then the height of the pole is

Options

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

Answer

To find the height of the pole, we can use the properties of right-angled triangles and trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the problem
We have a pole (AB) standing erect on the ground, and the tip of the pole is tied to a point on the ground (C) with a rope (AC) of length \( \sqrt{12} \) meters. The angle between the rope and the horizontal ground is \( 30^\circ \).

### Step 2: Identify the triangle
In the right triangle ABC:
- AB is the height of the pole (which we need to find).
- AC is the length of the rope (hypotenuse), which is \( \sqrt{12} \) meters.
- Angle ACB is \( 30^\circ \).

### Step 3: Use the sine function
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Thus, we can write:
\[ \sin(30^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} \]
Substituting the known values:
\[ \sin(30^\circ) = \frac{H}{\sqrt{12}} \]

### Step 4: Substitute the value of \( \sin(30^\circ) \)
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
So, we can rewrite the equation as:
\[ \frac{1}{2} = \frac{H}{\sqrt{12}} \]

### Step 5: Solve for H
To find H, we can cross-multiply:
\[ H = \frac{1}{2} \cdot \sqrt{12} \]
Now, simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]
Substituting this back into the equation:
\[ H = \frac{1}{2} \cdot 2\sqrt{3} = \sqrt{3} \]

### Step 6: Conclusion
Thus, the height of the pole is:
\[ H = \sqrt{3} \text{ meters} \]