The upper part of a tree broken over by wind, makes an angle of `30^(@)` with the ground. If the length of the upper part is `8sqrt3` metresthen what was the height of the tree before broken ?

Answer

To find the height of the tree before it was broken, we can follow these steps:

### Step 1: Understand the problem
We have a tree that was broken by the wind, and the upper part now makes an angle of \(30^\circ\) with the ground. The length of the broken part of the tree is \(8\sqrt{3}\) meters. We need to find the original height of the tree.

### Step 2: Draw a diagram
Let's visualize the situation:
- Let point A be the top of the tree before it was broken.
- Point B is where the tree is broken.
- Point C is the point on the ground directly below point A.
- The angle \( \angle ABC = 30^\circ \) and the length of \( AB = 8\sqrt{3} \) meters.

### Step 3: Identify the components of the triangle
In triangle ABC:
- \( AC \) is the height of the tree before it was broken.
- \( AB \) is the length of the broken part of the tree, which is \( 8\sqrt{3} \).
- \( BC \) is the horizontal distance from point B to point C.

### Step 4: Use the sine function
Using the sine function in triangle ABC:
\[ \sin(30^\circ) = \frac{AB}{AC} \]
Substituting the known values:
\[ \sin(30^\circ) = \frac{8\sqrt{3}}{AC} \]
We know that \( \sin(30^\circ) = \frac{1}{2} \), so we can write:
\[ \frac{1}{2} = \frac{8\sqrt{3}}{AC} \]

### Step 5: Solve for AC
Cross-multiplying gives:
\[ AC = 8\sqrt{3} \times 2 = 16\sqrt{3} \]

### Step 6: Calculate the total height of the tree
The total height of the tree before it was broken is the sum of \( AB \) and \( AC \):
\[ \text{Total height} = AC + AB = 16\sqrt{3} + 8\sqrt{3} = 24\sqrt{3} \text{ meters} \]

### Final Answer
The height of the tree before it was broken is \( 24\sqrt{3} \) meters.

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