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The problem describes two tasks where people earn ribbons:

1. **Total hours constraint**: A person may not serve more than 10 hours per week across both tasks.
2. **Minimum hours constraint**: A person must serve at least 1 hour per week on each task.

Let:
- \( k \) represent the number of hours a person spends helping at the food kitchen per week.
- \( t \) represent the number of hours a person spends planting trees per week.

### Constraints:
- The total hours spent on both tasks must be less than or equal to 10: \( k + t \leq 10 \).
- The time spent on each task must be at least 1 hour: \( k \geq 1 \) and \( t \geq 1 \).

### Matching with options:
- The correct system of inequalities that represents the constraints is:
  \[
  \begin{cases}
  k + t \leq 10 \\
  k \geq 1 \\
  t \geq 1
  \end{cases}
  \]

This matches **option C**.

Would you like more details or have any questions? Here are 5 related questions to explore further:

1. What would change if the maximum hours allowed per week was increased to 15?
2. How would the system change if there were no minimum hour requirements for either task?
3. What if the minimum hours for helping out at the food kitchen increased to 2 hours?
4. How could you represent this system graphically?
5. What other real-life scenarios might use similar systems of inequalities?

**Tip:** When solving inequality systems, make sure to break down each constraint and check if they fit the problem requirements individually.

At the Community Day event, people earn ribbons for two tasks: helping out at the food kitchen and planting trees at the park. The constraints are: A person may not serve more than 10 hours in total per week, and a person must serve at least 1 hour at each task. Which system of inequalities represents these constraints?

This problem involves setting up a system of inequalities to represent constraints on the total number of hours a person can serve at two tasks during a Community Day event. It covers key algebraic concepts such as inequalities and constraint-based problem solving, suitable for Grades 8-10.

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