Scenario:

Suppose a highly contagious virus spreads through a population. Initially, there are 10 people infected. The number of infected people doubles every 3 days due to the high transmission rate of the virus. We want to model the number of infected individuals over time and predict how many people will be infected after 15 days.

Mathematical Representation:

The situation can be modeled by an exponential growth function, which takes the form:

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

Where:

• $N(t)$ is the number of infected people at time $t$ (in days).
• $N_0 = 10$ is the initial number of infected people.
• $k$ is the growth rate constant.
• $t$ is the time in days.

Step 1: Finding the Growth Rate $k$

We know the number of infected people doubles every 3 days, which means:

$N(3) = 2 \cdot N_0$

Substituting into the exponential growth equation:

$2 \cdot 10 = 10 \cdot e^{3k}$

Simplifying:

$2 = e^{3k}$

Taking the natural logarithm (ln) of both sides:

$\ln(2) = 3k$

Solving for $k$:

$k = \frac{\ln(2)}{3}$

Using $\ln(2) \approx 0.693$, we get:

$k \approx \frac{0.693}{3} \approx 0.231$

Step 2: Predicting the Number of Infected People after 15 Days

Now that we have the growth rate $k$, we can use the exponential growth function to find the number of infected people after $t = 15$ days:

$N(15) = 10 \cdot e^{0.231 \cdot 15}$

Calculating the exponent:

$N(15) = 10 \cdot e^{3.465}$

$N(15) \approx 10 \cdot 32.03$

$N(15) \approx 320.3$

So, after 15 days, approximately 320 people will be infected.

Conclusion:

In this scenario, the number of infected people grows exponentially, and after 15 days, around 320 individuals will be infected with the virus.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How would the model change if the doubling time increased to 5 days?
2. What would happen if we introduced a treatment that reduced the infection rate?
3. How could this model be applied to predict the total population that will eventually be infected?
4. What if the initial number of infected people was different, say 50 instead of 10?
5. How does the assumption of continuous exponential growth differ from a real-world disease spread scenario?

Tip: When modeling exponential growth, doubling time is key to determining the growth rate, which helps in predicting future outcomes.