During the early stages of an epidemic, the number of people that fall ill doubles every 15 days.  Given this, what is the per-day rate of growth of the number of people that fall ill? 

To determine the per-day rate of growth during the early stages of an epidemic, we start by modeling the situation using exponential growth.

We are told that the number of people who fall ill doubles every 15 days. This means that, after 15 days, the population of infected individuals has grown by a factor of 2.

The exponential growth formula is:

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

Where:
- \( N(t) \) is the number of people infected at time \( t \),
- \( N_0 \) is the initial number of infected people,
- \( r \) is the growth rate (per day),
- \( t \) is the time in days.

### Step 1: Relating the given doubling time to the growth rate
After 15 days, the population has doubled, so \( N(15) = 2N_0 \). Using the exponential model:

\[
2N_0 = N_0 \cdot e^{r \cdot 15}
\]

### Step 2: Solving for \( r \)
Divide both sides by \( N_0 \):

\[
2 = e^{15r}
\]

Take the natural logarithm of both sides to isolate \( r \):

\[
\ln(2) = 15r
\]

Now solve for \( r \):

\[
r = \frac{\ln(2)}{15} \approx \frac{0.6931}{15} \approx 0.0462
\]

Thus, the per-day growth rate \( r \) is approximately \( 0.0462 \), or about **4.62% per day**.

Would you like more details, or do you have any questions?

Here are 5 follow-up questions related to this concept:
1. How does the growth rate change if the doubling time is shortened?
2. What happens to the population size after 30 days with the same growth rate?
3. How can we adjust this model for epidemics with a variable growth rate?
4. What real-world factors might slow down or accelerate this type of exponential growth?
5. How would you calculate the total number of people infected after a specific number of days?

**Tip**: Always double-check your exponential growth assumptions, as they can shift due to external factors like intervention measures.

During the early stages of an epidemic, the number of people that fall ill doubles every 15 days. Given this, what is the per-day rate of growth of the number of people that fall ill?

Understand how to calculate the per-day growth rate during the early stages of an epidemic where the number of cases doubles every 15 days. This problem uses exponential growth models and logarithmic functions, suitable for students in Grades 9-12.

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