Consider this scenario: The number of people afflicted with the common cold *C* in the winter months steadily decreased by 206 each year from 2005 until 2010. In 2005, 12,065 people were afflicted. Consider the linear function that models the number of people afflicted with the common cold *C* as a function of the year *t*. (Let *t* represent the number of years since 2005.)

Find a reasonable domain for the function *C*.

−1,030 ≤ *C* ≤ 0

11,035 ≤ *C* ≤ 12,065

11,035 ≤ *t* ≤ 12,065

0 ≤ *C* ≤ 5

0 ≤ *t* ≤ 5

Find a reasonable range for the function *C*.

−1,030 ≤ *C* ≤ 0

11,035 ≤ *C* ≤ 12,065

11,035 ≤ *t* ≤ 12,065

0 ≤ *C* ≤ 5

0 ≤ *t* ≤ 5

To find a reasonable domain and range for the function \( C(t) \) that models the number of people afflicted with the common cold, we'll go step by step.

### 1. Understanding the Linear Function \( C(t) \)
Given that the number of people afflicted decreases by 206 each year, we can express \( C(t) \) as:
\[
C(t) = 12065 - 206t
\]
where:
- \( t \) is the number of years since 2005.
- \( C(t) \) is the number of people afflicted with the cold in year \( t \).

### 2. Reasonable Domain for \( t \)
Since \( t \) represents the number of years since 2005 and the time period in question is from 2005 to 2010, \( t \) ranges from 0 (representing 2005) to 5 (representing 2010). Therefore, the reasonable domain for \( t \) is:
\[
0 \leq t \leq 5
\]
So, the correct domain is:
\[
0 \leq t \leq 5
\]

### 3. Reasonable Range for \( C(t) \)
Next, let's find the values of \( C(t) \) at the endpoints of the domain:

- When \( t = 0 \) (in 2005):
  \[
  C(0) = 12065 - 206(0) = 12065
  \]

- When \( t = 5 \) (in 2010):
  \[
  C(5) = 12065 - 206(5) = 12065 - 1030 = 11035
  \]

Thus, the number of people afflicted, \( C(t) \), ranges from 11,035 to 12,065. Therefore, the reasonable range for \( C(t) \) is:
\[
11035 \leq C(t) \leq 12065
\]

### Final Answer:

- **Reasonable Domain for \( t \):** \( 0 \leq t \leq 5 \)
- **Reasonable Range for \( C(t) \):** \( 11035 \leq C(t) \leq 12065 \)

These match the answer options:

- **Domain:** \( 0 \leq t \leq 5 \)
- **Range:** \( 11035 \leq C \leq 12065 \)

Would you like further details on how the linear function was determined, or do you have any other questions?

Here are some related questions you might find helpful:
1. What is the slope of the linear function \( C(t) \)?
2. How do you interpret the y-intercept in the context of this problem?
3. What would the function look like if the number of afflicted people increased instead?
4. How would the range change if the period extended to 2015?
5. How could you verify if this linear model is accurate for years beyond 2010?

**Tip:** Understanding the relationship between domain and range is crucial for interpreting real-world scenarios in function models.

Consider this scenario: The number of people afflicted with the common cold C in the winter months steadily decreased by 206 each year from 2005 until 2010. In 2005, 12,065 people were afflicted. Consider the linear function that models the number of people afflicted with the common cold C as a function of the year t. (Let t represent the number of years since 2005.)

Find a reasonable domain for the function C.

−1,030 ≤ C ≤ 0

11,035 ≤ C ≤ 12,065

11,035 ≤ t ≤ 12,065

0 ≤ C ≤ 5

0 ≤ t ≤ 5

Find a reasonable range for the function C.

−1,030 ≤ C ≤ 0

11,035 ≤ C ≤ 12,065

11,035 ≤ t ≤ 12,065

0 ≤ C ≤ 5

0 ≤ t ≤ 5

Learn how to determine the domain and range of a linear function modeling the number of people afflicted with the common cold from 2005 to 2010. This problem involves understanding linear equations, calculating endpoints, and interpreting real-world data. Suitable for students in Grades 8-10.

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