In a violent storm, a tree got bent by the wind. The top of the tree meets the ground at an angle of `30^(@)`, at a distance of 30 metres from the root. At what height from the bottom did the tree bent? What was the original height of the tree? (Use `sqrt3=1.73`)

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a tree that has bent at a certain height due to a storm. The top of the tree meets the ground at an angle of \(30^\circ\) and is \(30\) meters away from the root of the tree. We need to find out at what height from the bottom the tree bent and what the original height of the tree was.

### Step 2: Define the Points
Let:
- \(A\) be the point where the tree bends.
- \(B\) be the top of the tree.
- \(C\) be the point on the ground directly below \(B\).
- \(O\) be the root of the tree.

### Step 3: Identify the Triangle
In triangle \(ABC\):
- \(AB\) is the part of the tree that remains standing.
- \(BC\) is the distance from the root to the point where the top of the tree meets the ground, which is given as \(30\) meters.
- The angle \(CAB\) is \(30^\circ\).

### Step 4: Use Trigonometric Ratios
Using the tangent function in triangle \(ABC\):
\[ \tan(30^\circ) = \frac{AB}{BC} \]
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Substituting the known values:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{30} \]

### Step 5: Solve for \(AB\)
Rearranging the equation gives:
\[ AB = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} = 10\sqrt{3} \]
Using the approximation \(\sqrt{3} \approx 1.73\):
\[ AB \approx 10 \cdot 1.73 = 17.3 \text{ meters} \]

### Step 6: Find \(AC\) Using Sine
Now, we will find the original height \(AC\) using the sine function:
\[ \sin(30^\circ) = \frac{AB}{AC} \]
Since \(\sin(30^\circ) = \frac{1}{2}\):
\[ \frac{1}{2} = \frac{10\sqrt{3}}{AC} \]
Rearranging gives:
\[ AC = 2 \cdot 10\sqrt{3} = 20\sqrt{3} \]
Using the approximation:
\[ AC \approx 20 \cdot 1.73 = 34.6 \text{ meters} \]

### Step 7: Calculate the Original Height of the Tree
The total height of the tree \(h\) is the sum of \(AB\) and \(AC\):
\[ h = AB + AC = 10\sqrt{3} + 20\sqrt{3} = 30\sqrt{3} \]
Using the approximation:
\[ h \approx 30 \cdot 1.73 = 51.9 \text{ meters} \]

### Final Answers
1. The height from the bottom where the tree bent is approximately \(17.3\) meters.
2. The original height of the tree is approximately \(51.9\) meters.