A portion of a 30 m 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. `30sqrt3m`
B. 10 m
C. `30sqrt3m`
D. 60 m

Answer

To find the height of the point where the tree is broken, we can use the properties of a right triangle formed by the broken tree. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a tree that is 30 meters long, which has broken and fallen to the ground, making an angle of 30 degrees with the ground. We need to find the height at which the tree broke.

### Step 2: Visualize the Situation
When the tree breaks, it forms a right triangle where:
- The length of the tree (hypotenuse) = 30 meters.
- The angle with the ground = 30 degrees.

Let:
- Point A = the top of the tree (where it broke).
- Point B = the point on the ground directly below A.
- Point C = the point where the tree is broken.

### Step 3: Identify the Sides of the Triangle
In triangle ABC:
- AB (the height from the ground to the break point) is opposite the angle of 30 degrees.
- BC (the horizontal distance from the break point to the point directly below) is adjacent to the angle of 30 degrees.
- AC (the length of the tree) is the hypotenuse = 30 meters.

### Step 4: Use Trigonometric Ratios
Using the sine function, which relates the opposite side to the hypotenuse:
\[ \sin(30^\circ) = \frac{AB}{AC} \]
Substituting the known values:
\[ \sin(30^\circ) = \frac{AB}{30} \]
We know that \(\sin(30^\circ) = \frac{1}{2}\):
\[ \frac{1}{2} = \frac{AB}{30} \]

### Step 5: Solve for AB
To find AB (the height):
\[ AB = 30 \times \frac{1}{2} = 15 \text{ meters} \]

### Step 6: Conclusion
The height at which the tree is broken is 15 meters.

### Final Answer
The height of the point where the tree is broken is **15 meters**.