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.