Gauri's father wanted to build a compound wall in her plot. For that, she was supposed to cut a tree at the boundary. To cut the tree she fastened a rope in a big rock on to the top of the tree at an angle of 60∘ from the ground. First, she cut at a point D on the tree such that the tree bends and touched the ground at a distance of 7.2 m from the base of the tree making an angle of 60∘. Find the distance of the rock to the point where the fallen wooden log met the ground

To solve this problem, we need to break it down using basic trigonometry, as the situation involves angles and distances in a right-angled triangle.

### Step-by-step process:

1. **Visualizing the setup**:
    - The tree was cut and bent at an angle of \(60^\circ\), falling to the ground 7.2 meters from the base of the tree.
    - A rope was tied from the top of the tree to a rock on the ground, forming another right-angled triangle where the rope makes an angle of \(60^\circ\) with the ground.

2. **Given Information**:
    - The angle at which the tree bent and fell to the ground is \(60^\circ\).
    - The distance from the base of the tree to the point where the tree touches the ground is 7.2 meters.
    - The angle between the rope and the ground is \(60^\circ\).

3. **Objective**:
    We are required to find the distance from the rock to the point where the fallen log meets the ground.

4. **Using trigonometry**:
    In this scenario, the rope forms a right-angled triangle with the ground. We can use the sine function to calculate the length of the rope.

    In a right-angled triangle:
    \[
    \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
    \]
    Here, the opposite side is the distance from the base of the tree to the point where the tree touches the ground (7.2 meters), the angle is \(60^\circ\), and the hypotenuse is the length of the rope (which we want to find).

5. **Formula application**:
    We can rearrange the formula to solve for the hypotenuse (the length of the rope):
    \[
    \text{Hypotenuse} = \frac{\text{Opposite}}{\sin(\theta)}
    \]
    Substituting the known values:
    \[
    \text{Length of the rope} = \frac{7.2}{\sin(60^\circ)}
    \]

6. **Calculation**:
    Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866\),
    \[
    \text{Length of the rope} = \frac{7.2}{0.866} \approx 8.31 \text{ meters}.
    \]

### Conclusion:
The distance from the rock to the point where the fallen tree touches the ground is approximately **8.31 meters**.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the height of the tree before it was cut?
2. How would the result change if the tree had fallen at a different angle?
3. Can you calculate the height of the tree after it was bent?
4. What is the length of the rope if the angle of inclination changes to \(45^\circ\)?
5. How do different trigonometric functions relate to real-world applications like this?

**Tip**: Always draw a diagram for problems involving angles and distances to get a clear visual understanding of the situation.

Gauri's father wanted to build a compound wall in her plot. For that, she was supposed to cut a tree at the boundary. To cut the tree she fastened a rope in a big rock on to the top of the tree at an angle of 60∘ from the ground. First, she cut at a point D on the tree such that the tree bends and touched the ground at a distance of 7.2 m from the base of the tree making an angle of 60∘. Find the distance of the rock to the point where the fallen wooden log met the ground.

Solve a real-world trigonometry problem involving a tree cut and a rope tied at a 60-degree angle. Learn how to calculate the distance from a rock to the fallen log using the sine function, with detailed step-by-step explanations. Suitable for students in Grades 8-10.

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