Diseases tend to spread according to the exponential growth model. In the early days of AIDS, the growth factor (i.e. common ratio; growth multiplier) was around 2.0. In 1983, about 1800 people in the U.S. died of AIDS. If the trend had continued unchecked, how many people would have died from AIDS in 2005? x people (Note: once diseases become widespread, they start to behave more like logistic growth, but don't worry about that for the purpose of this exercise)

We can model the number of AIDS deaths using an exponential growth formula:

\[
P(t) = P_0 \cdot r^t
\]

where:
- \( P(t) \) is the population (or number of deaths) at time \( t \),
- \( P_0 \) is the initial population (number of deaths at the starting year),
- \( r \) is the growth factor (or common ratio),
- \( t \) is the number of years since the initial year.

### Given values:
- Initial year: 1983, where 1800 people died, so \( P_0 = 1800 \),
- Growth factor \( r = 2.0 \),
- We want to calculate the number of deaths in 2005, so \( t = 2005 - 1983 = 22 \) years.

Now, applying these values into the exponential growth formula:

\[
P(22) = 1800 \cdot 2^{22}
\]

To calculate this:

\[
P(22) = 1800 \cdot 4194304 = 7,549,747,200
\]

Thus, if the trend had continued unchecked, approximately **7.55 billion** people would have died from AIDS in 2005.

Would you like more details or have any questions?

Here are 5 related questions to further explore the concept:
1. How does exponential growth differ from logistic growth in disease modeling?
2. What factors might cause an exponential growth rate to slow down over time?
3. How would a lower growth factor (e.g., 1.5 instead of 2.0) have affected the result?
4. How would the prediction change if we considered the logistic growth model?
5. What real-world interventions can significantly alter disease growth patterns?

**Tip:** Exponential growth models often apply to early-stage epidemics, but eventually, factors like herd immunity, medical interventions, and resource limitations shift the model towards logistic growth.

Explore the exponential growth model applied to the spread of diseases, using AIDS as a case study. Given a growth factor of 2.0 and 1800 deaths in 1983, discover how many people would have died by 2005 if the trend had continued unchecked. Perfect for Grades 9-12, this problem highlights key concepts in epidemiology and exponential growth.

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