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The problem presents a function \( d(t) = 6.6t \) that models the distance \( d \) in kilometers Kellen runs, where \( t \) is the time in hours. The given conditions specify that Kellen runs for at least 1 hour but no more than 2 hours.

### To determine the domains:
- **Theoretical Domain**: The theoretical domain refers to all possible values of \( t \) that the function could take, regardless of real-world constraints. Since \( d(t) = 6.6t \) is a linear function with no restrictions based on the equation itself, the theoretical domain could be **all real numbers** (i.e., \( t \in (-\infty, \infty) \)).

- **Practical Domain**: The practical domain refers to the real-world constraints given in the problem. Kellen runs for a time \( t \) between 1 hour and 2 hours, inclusive. So, the practical domain is **[1, 2]**, meaning \( t \) can be any real number between 1 and 2, inclusive.

### Correct answers:
- The theoretical domain is all real numbers.
- The practical domain is all real numbers between 1 and 2, inclusive.

Would you like further clarification or details on this?

Here are 5 related questions to explore this topic further:
1. How would the range of the function change if Kellen’s running speed was different?
2. What is the distance Kellen runs if he runs for exactly 1.5 hours?
3. How would you graph the function \( d(t) = 6.6t \) on a coordinate plane?
4. What would the practical domain be if the time limit changed to 0.5 hours to 3 hours?
5. How do theoretical and practical domains differ in real-world problems?

**Tip**: Always check for any real-world constraints when determining the practical domain in applied mathematics problems.

Kellen runs for at least 1 h but for no more than 2 h. He runs at an average rate of 6.6 km/h. The function that models the distance he runs for t hours is d(t) = 6.6t. What are the theoretical and practical domains of the function?

Learn how to determine the theoretical and practical domains of the function d(t) = 6.6t, which models the distance Kellen runs between 1 and 2 hours. This problem covers key concepts of linear functions and function domains, suitable for Grades 9-11.

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