In a storm, a tree got bent by the wind whose top meets the ground at an angle of `30^(@)`, at a distance of 30 meters from the root. What is the height of the tree.

Answer

To find the height of the tree that got bent by the wind, we can use trigonometric principles. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
When the tree bends, it forms a right triangle where:
- The height of the tree (h) is one leg of the triangle.
- The distance from the root to the point where the top of the tree touches the ground (30 meters) is the other leg.
- The angle between the ground and the line from the top of the tree to the point on the ground is 30 degrees.

### Step 2: Identify the Angles
In the right triangle:
- The angle at the top of the tree is 30 degrees.
- The angle at the base (where the tree meets the ground) is 90 degrees.
- The remaining angle must be 60 degrees (since the angles in a triangle sum up to 180 degrees).

### Step 3: Use Trigonometric Ratios
Using the tangent function, which is defined as the opposite side over the adjacent side in a right triangle:
\[ \tan(30^\circ) = \frac{\text{height of the tree (h)}}{\text{distance from the root (30 m)}} \]

### Step 4: Calculate the Height
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \text{ or } \frac{\sqrt{3}}{3} \]
Thus, we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h}{30} \]

Now, solving for h:
\[ h = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ meters} \]

### Step 5: Total Height of the Tree
Since the tree is bent, the total height of the tree will be the height above the ground plus the height of the bent part of the tree. The height of the bent part can be calculated using the sine function:
\[ \sin(30^\circ) = \frac{\text{height of the bent part}}{30} \]
Since \(\sin(30^\circ) = \frac{1}{2}\):
\[ \frac{1}{2} = \frac{\text{height of the bent part}}{30} \]
Thus, the height of the bent part is:
\[ \text{height of the bent part} = 30 \cdot \frac{1}{2} = 15 \text{ meters} \]

### Step 6: Add the Heights
Now, we can find the total height of the tree:
\[ \text{Total height} = h + \text{height of the bent part} = 10\sqrt{3} + 15 \]

### Final Answer
The total height of the tree is:
\[ 10\sqrt{3} + 15 \text{ meters} \]