A tree breaks due to storm and the broken part bends, so that the top of the tree touches the ground making an angle of `60^(@)` with the ground . The distance from the foot of the tree to the point where the top touches the ground is 5 metres . Find the height of the tree. (`sqrt(3) = 1.73`)

Answer

To solve the problem, we need to find the height of the tree after it has broken and bent down to touch the ground at an angle of \(60^\circ\). We know the distance from the foot of the tree to the point where the top touches the ground is 5 meters.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- When the tree breaks and bends, it forms a right triangle where:
- The distance from the foot of the tree to the point where the top touches the ground is the base (adjacent side) of the triangle, which is 5 meters.
- The height of the tree before it broke is the perpendicular side (opposite side) of the triangle.
- The broken part of the tree forms the hypotenuse.

2. **Identifying the Angles**:
- The angle between the ground and the broken part of the tree is \(60^\circ\).

3. **Using Cosine to Find the Hypotenuse**:
- We can use the cosine function to find the length of the hypotenuse (the broken part of the tree).
- The formula for cosine is:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
- For our triangle:
\[ \cos(60^\circ) = \frac{5}{\text{Hypotenuse}} \]
- We know that \(\cos(60^\circ) = \frac{1}{2}\), so:
\[ \frac{1}{2} = \frac{5}{\text{Hypotenuse}} \]
- Rearranging gives:
\[ \text{Hypotenuse} = 5 \times 2 = 10 \text{ meters} \]

4. **Using Sine to Find the Height**:
- Now, we can use the sine function to find the height (perpendicular side).
- The formula for sine is:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
- For our triangle:
\[ \sin(60^\circ) = \frac{\text{Height}}{10} \]
- We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), so:
\[ \frac{\sqrt{3}}{2} = \frac{\text{Height}}{10} \]
- Rearranging gives:
\[ \text{Height} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ meters} \]

5. **Calculating the Height**:
- We can substitute \(\sqrt{3} \approx 1.73\):
\[ \text{Height} \approx 5 \times 1.73 = 8.65 \text{ meters} \]

6. **Total Height of the Tree**:
- The total height of the tree before it broke is the height of the broken part plus the height of the part that remains upright (which is also the height we calculated):
\[ \text{Total Height} = \text{Height} + \text{Height of the upright part} \]
- Since the tree was originally standing upright, the total height of the tree is:
\[ \text{Total Height} = 8.65 + 10 = 18.65 \text{ meters} \]

### Final Answer:
The height of the tree is approximately **18.65 meters**.