If the number of infections starts at 8 and continues to grow exponentially, what will the number of infections be in week 7?

To determine the number of infections in week 7 given exponential growth, we use the formula:

\[
N(t) = N_0 \cdot e^{kt}
\]

where:
- \( N_0 \) is the initial number of infections (8 in this case),
- \( k \) is the growth rate constant,
- \( t \) is the time in weeks, and
- \( N(t) \) is the number of infections at time \( t \).

Without a specific value for \( k \), we can't calculate the exact number. However, if we assume the infections double each week (a common exponential growth scenario), we can use:

\[
N(t) = N_0 \cdot 2^t
\]

### Calculation with Doubling Assumption
For this scenario:
- Initial infections, \( N_0 = 8 \),
- Weeks, \( t = 7 \).

So,

\[
N(7) = 8 \cdot 2^7 = 8 \cdot 128 = 1024
\]

Therefore, the number of infections in week 7 would be **1024** if the infections double every week.

Would you like further details or have any questions?

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### Follow-up Questions
1. What is the growth rate constant \( k \) if infections double weekly?
2. How would the infection count change if the growth rate were slower, say 1.5 times each week?
3. How would we modify the model if the growth rate were given as a daily percentage?
4. What would the model look like for a linear, rather than exponential, increase?
5. How does an exponential growth model apply to real-life epidemics?

**Tip:** In exponential growth, small changes in the growth rate can lead to significant differences in outcomes over time.

Learn how to calculate the number of infections in week 7 given exponential growth, assuming infections double every week. Using key formulas such as N(t) = N_0 * 2^t, this problem highlights the principles of exponential growth, perfect for students in Grades 9-12.

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