To find the linear function that models the number of people afflicted with the common cold $C(t)$ as a function of the year $t$, we can use the following steps:

Step 1: Determine the slope (rate of change)

The problem states that the number of people afflicted decreases by 25 each year. This means the slope $m$ of the linear function is:

$m = -25$

Step 2: Find the y-intercept

In 2002 (which we can take as $t = 0$), the number of people afflicted was 8,040. The linear function can be expressed in the form:

$C(t) = mt + C_0$

Where $C_0$ is the initial value (y-intercept).

Given:

• $C(0) = 8040$
• $m = -25$

Thus, the linear function is:

$C(t) = -25t + 8040$

Step 3: Determine when the number of people afflicted drops below 6,000

We want to find the year $t$ when $C(t) < 6000$. So, set the function equal to 6,000 and solve for $t$:

$6000 = -25t + 8040$

Subtract 8040 from both sides:

$6000 - 8040 = -25t$

$-2040 = -25t$

Divide both sides by -25:

$t = \frac{-2040}{-25} = 81.6$

Since $t = 0$ corresponds to the year 2002, $t = 81.6$ corresponds to the year $2002 + 81.6 = 2083.6$.

Therefore, the number of people afflicted will drop below 6,000 sometime in the year 2083.

Final Answer

• The linear function is $C(t) = -25t + 8040$.
• The number of people afflicted with the common cold will drop below 6,000 in the year 2083.

Would you like more details or have any questions?

Here are some related questions:

1. What does the slope of the linear function represent in real life?
2. How would the function change if the rate of decrease was different?
3. What if the initial number of people afflicted was 10,000 in 2002?
4. How can we model the situation if the decline wasn't steady but exponential?
5. What does the y-intercept of the function tell us in this context?

Tip: When interpreting a linear function, the slope indicates the rate of change, and the y-intercept represents the initial value when the independent variable (like time) is zero.