To find the length of time it takes for the number of infected people to reach 1,000,000, we use the same exponential growth model $A(t) = I e^{rt}$ and solve for $t$.

Given:

• Initial number of infected people, $I = 28$
• Growth rate, $r = 0.045$
• Final number of infected people, $A(t) = 1,000,000$

We need to solve for $t$: $1,000,000 = 28 e^{0.045t}$

First, divide both sides by 28 to isolate the exponential term: $\frac{1,000,000}{28} = e^{0.045t}$ $35,714.2857 \approx e^{0.045t}$

Next, take the natural logarithm (ln) of both sides to solve for $t$: $\ln(35,714.2857) = \ln(e^{0.045t})$ $\ln(35,714.2857) = 0.045t$

Calculate the natural logarithm of 35,714.2857: $\ln(35,714.2857) \approx 10.48$

Now, solve for $t$: $10.48 = 0.045t$ $t = \frac{10.48}{0.045}$ $t \approx 232.89$

Rounding to the nearest whole number: $t \approx 233$

So, it takes approximately 233 days for the number of infected people to reach 1,000,000.

Would you like any further details or have any questions?

Here are 5 more questions you might want to ask:

1. How does the doubling time of an infection relate to the growth rate?
2. How would the model change if we had a carrying capacity for the population?
3. What would happen if the growth rate decreased over time?
4. Can you show how to solve the same problem using a different method?
5. How can this model be used to predict the future course of an epidemic?

Tip: When working with exponential growth models, always check if the assumptions (constant growth rate and initial population) are valid over the time period you're examining.