n 20082008 there were 7070 million people worldwide who had been infected with a particular disease. At that time the infection rate was 4.74.7 million people per year.
​(a) Write a formula for a linear function that models the total number of people in millions who were infected with the particular disease x years after 20082008.
​(b) Estimate the number of people who may have been infected by the year 20132013.


Let's break down this problem into parts.

### Part (a): Formula for the linear function

We are given:
- In 2008, 70 million people were infected.
- The infection rate is 4.7 million people per year.

We need to write a linear function that models the total number of people infected \( y(x) \), where \( x \) represents the number of years after 2008.

The general form of a linear function is:
\[
y(x) = mx + b
\]
Where:
- \( m \) is the slope (rate of change), which is the infection rate (4.7 million people per year),
- \( b \) is the y-intercept, which is the initial value (70 million people in 2008, or when \( x = 0 \)).

Thus, the formula for the linear function is:
\[
y(x) = 4.7x + 70
\]
This equation models the total number of people infected \( x \) years after 2008.

---

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

To estimate the number of people infected by the year 2013, we need to find the value of \( y(x) \) when \( x = 2013 - 2008 = 5 \).

Using the function \( y(x) = 4.7x + 70 \), plug in \( x = 5 \):
\[
y(5) = 4.7(5) + 70
\]
\[
y(5) = 23.5 + 70 = 93.5
\]
So, the estimated number of people infected by the year 2013 is **93.5 million**.

---

### Summary:
- The linear function is \( y(x) = 4.7x + 70 \).
- The estimated number of people infected by 2013 is 93.5 million.

---

Do you need further details or have any questions?

Here are 5 related questions to explore:
1. How does the infection rate change if the number of new cases increases by 1 million per year?
2. What will be the estimated number of infections in 2020 using the same model?
3. If the infection rate decreases to 3 million per year after 2015, how would the formula change?
4. What is the significance of the slope and intercept in this context?
5. How can this model be adjusted for future trends or other factors like vaccination rates?

**Tip**: Linear models are good for estimating trends over short periods, but real-world data might require more complex models for long-term accuracy!

In 2008, there were 70 million people worldwide who had been infected with a particular disease. At that time, the infection rate was 4.7 million people per year. (a) Write a formula for a linear function that models the total number of people in millions who were infected with the disease x years after 2008. (b) Estimate the number of people who may have been infected by the year 2013.

This problem explores how to write a linear function to model the number of people infected by a disease over time, given an initial infection count of 70 million and an annual rate of 4.7 million people. The solution provides a step-by-step explanation of the formula y(x) = 4.7x + 70 and estimates the total infections by 2013.

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