some portion of a 30m long tree is broken by tornado and the top struck up the ground making an angle `30^@` with ground level. The height of the point where the tree is broken is equal to:

Options

A. `(30)/sqrt3` m
B. 10 m
C. `30sqrt3` m
D. 60 m

Answer

To solve the problem, we can use trigonometry. We know that the total length of the tree is 30 meters and that the top of the tree makes an angle of 30 degrees with the ground after it has broken. We need to find the height at which the tree broke.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The tree is originally 30 meters long.
- After the tornado, the top of the tree touches the ground, forming an angle of 30 degrees with the ground.

2. **Setting Up the Triangle**:
- Let \( x \) be the height from the ground to the point where the tree broke.
- The remaining part of the tree that is lying on the ground will be \( 30 - x \).
- We can visualize this scenario as a right triangle where:
- The height \( x \) is the opposite side.
- The length of the tree that is on the ground \( 30 - x \) is the adjacent side.
- The hypotenuse is the original length of the tree, which is 30 meters.

3. **Using the Sine Function**:
- According to the sine function in a right triangle:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
- Here, \( \theta = 30^\circ \), the opposite side is \( x \), and the hypotenuse is \( 30 \).
- Therefore, we can write:
\[ \sin(30^\circ) = \frac{x}{30 - x} \]

4. **Substituting the Value of Sine**:
- We know that \( \sin(30^\circ) = \frac{1}{2} \).
- Substituting this into the equation gives:
\[ \frac{1}{2} = \frac{x}{30 - x} \]

5. **Cross-Multiplying**:
- Cross-multiplying to eliminate the fraction:
\[ 1 \cdot (30 - x) = 2x \]
- This simplifies to:
\[ 30 - x = 2x \]

6. **Solving for x**:
- Rearranging the equation:
\[ 30 = 2x + x \]
\[ 30 = 3x \]
- Dividing both sides by 3 gives:
\[ x = 10 \]

7. **Conclusion**:
- The height at which the tree is broken is \( x = 10 \) meters.

### Final Answer:
The height of the point where the tree is broken is **10 meters**.