A wire is tied to the top of a pole whose height is 15 meter and the wire makes an angle of 60° with the ground. The length of the wire is :

Question

Accepted Answer

To determine the length of the wire, we can use trigonometry. The wire forms a right triangle with the pole and the ground. The pole is the opposite side to the angle, the wire is the hypotenuse, and the ground is the adjacent side.

Given:
- Height of the pole (opposite side) = 15 meters
- Angle with the ground = 60°

We need to find the length of the wire (hypotenuse). We can use the sine function, which relates the opposite side and the hypotenuse in a right triangle:

$$
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
$$

Substituting the given values:

$$
\sin(60°) = \frac{15}{\text{hypotenuse}}
$$

We know that $\sin(60°) = \frac{\sqrt{3}}{2}$. Therefore:

$$
\frac{\sqrt{3}}{2} = \frac{15}{\text{hypotenuse}}
$$

Solving for the hypotenuse:

$$
\text{hypotenuse} = \frac{15 \times 2}{\sqrt{3}} = \frac{30}{\sqrt{3}}
$$

To rationalize the denominator:

$$
\text{hypotenuse} = \frac{30 \times \sqrt{3}}{3} = 10\sqrt{3} \text{ meters}
$$

Thus, the length of the wire is $10\sqrt{3}$ meters, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1: 30 meters** - This option is incorrect because it does not account for the trigonometric relationship between the height of the pole and the angle. The calculation using the sine function shows that the hypotenuse is not 30 meters.

2. **Option 2: $20\sqrt{3}$ meters** - This option is incorrect because it results from an incorrect calculation. The correct trigonometric calculation gives $10\sqrt{3}$ meters, not $20\sqrt{3}$ meters.

3. **Option 4: $30\sqrt{3}$ meters** - This option is incorrect as it significantly overestimates the length of the wire. The correct calculation using the sine function yields $10\sqrt{3}$ meters.

In conclusion, the correct length of the wire, based on the given angle and height, is $10\sqrt{3}$ meters, which is why Option 3 is the correct answer.