The upper part of a tree broken over by the wind makes an angle of `30^(@)` with the ground and the distance of the root from the point where the top touches the ground is 25 m. What was the total height of the tree?

Answer

To solve the problem step by step, we will use trigonometric ratios to find the height of the tree before it was broken.

### Step 1: Understand the problem
We have a tree that has been broken by the wind. The upper part of the tree makes an angle of \(30^\circ\) with the ground, and the distance from the root of the tree to the point where the top touches the ground is 25 meters.

### Step 2: Set up the triangle
We can visualize this situation as a right triangle:
- Let \(x\) be the height of the remaining part of the tree (the part that is still standing).
- Let \(y\) be the length of the broken part of the tree (the part that is tilted down).
- The distance from the root to the point where the top touches the ground is the base of the triangle, which is 25 meters.

### Step 3: Use the tangent function
Using the angle \(30^\circ\), we can apply the tangent function:
\[ \tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{25} \]
From trigonometric tables, we know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus, we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{x}{25} \]

### Step 4: Solve for \(x\)
To find \(x\), we can rearrange the equation:
\[ x = 25 \cdot \frac{1}{\sqrt{3}} = \frac{25}{\sqrt{3}} \text{ meters} \]

### Step 5: Use the cosine function to find \(y\)
Next, we will find \(y\) using the cosine function:
\[ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{25}{y} \]
From trigonometric tables, we know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Thus, we can set up the equation:
\[ \frac{\sqrt{3}}{2} = \frac{25}{y} \]

### Step 6: Solve for \(y\)
Rearranging the equation gives:
\[ y = \frac{25 \cdot 2}{\sqrt{3}} = \frac{50}{\sqrt{3}} \text{ meters} \]

### Step 7: Find the total height of the tree
The total height of the tree before it was broken is the sum of \(x\) and \(y\):
\[ \text{Total height} = x + y = \frac{25}{\sqrt{3}} + \frac{50}{\sqrt{3}} = \frac{75}{\sqrt{3}} \text{ meters} \]

### Step 8: Rationalize the answer
To express the answer in a more standard form, we rationalize:
\[ \frac{75}{\sqrt{3}} = \frac{75 \cdot \sqrt{3}}{3} = 25\sqrt{3} \text{ meters} \]

### Final Answer
The total height of the tree is \(25\sqrt{3}\) meters.

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