We can solve the problem in two parts, as requested:

Part (a): Write a formula for a linear function

In 2008, there were 70 million people infected, and the infection rate was increasing by 4.7 million people per year. This scenario can be modeled by a linear function.

Let:

• $P(x)$ represent the total number of infected people (in millions) $x$ years after 2008.
• $x$ represent the number of years after 2008.

The linear function can be written in the slope-intercept form:
$P(x) = mx + b$ Where:

• $m$ is the rate of change (slope), which is 4.7 million people per year.
• $b$ is the initial value in 2008, which is 70 million.

Thus, the formula for the linear function is: $P(x) = 4.7x + 70$

Part (b): Estimate the number of people infected by 2013

To estimate the number of people infected by 2013, we first calculate the value of $x$. Since 2013 is 5 years after 2008: $x = 2013 - 2008 = 5$ Now, substitute $x = 5$ into the linear function: $P(5) = 4.7(5) + 70 = 23.5 + 70 = 93.5$ Thus, the estimated number of people infected by the year 2013 is 93.5 million.

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Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the model change if the infection rate increased by 10% every year?
2. What would be the infection count in 2020 using this model?
3. How would the graph of this function look over a 20-year span?
4. How does the slope $m = 4.7$ affect the prediction for future years?
5. Can you express the infection rate as a percentage of the population per year?

Tip: In linear models, the slope represents the rate of change, giving us key insights into how quickly the situation evolves over time.