An epidemic has an exponential growth rate (k) of 0.32 per annum. Assuming an initial infected population of 3, how long before 800 people are infected

To determine how long it will take for the infected population to grow from 3 to 800 people in an epidemic with an exponential growth rate, we can use the exponential growth formula:

\[ P(t) = P_0 e^{kt} \]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the growth rate,
- \( t \) is the time.

Given:
- \( P_0 = 3 \),
- \( P(t) = 800 \),
- \( k = 0.32 \) per annum.

We need to find \( t \). Start by plugging the given values into the formula:

\[ 800 = 3 e^{0.32t} \]

Next, solve for \( t \):

1. Divide both sides by 3:
   \[ \frac{800}{3} = e^{0.32t} \]

2. Simplify the fraction:
   \[ 266.67 = e^{0.32t} \]

3. Take the natural logarithm of both sides to solve for \( t \):
   \[ \ln(266.67) = \ln(e^{0.32t}) \]

4. Use the property of logarithms that \( \ln(e^x) = x \):
   \[ \ln(266.67) = 0.32t \]

5. Solve for \( t \):
   \[ t = \frac{\ln(266.67)}{0.32} \]

6. Calculate the natural logarithm and division:
   \[ t \approx \frac{5.586}{0.32} \approx 17.46 \]

Therefore, it will take approximately 17.46 years for the infected population to grow from 3 to 800 people.

Would you like more details or have any questions?

Here are some related questions:
1. What is exponential growth and how is it different from linear growth?
2. How does the growth rate (k) affect the time it takes for the population to reach a certain size?
3. How can we determine the doubling time for an epidemic given the growth rate?
4. What other factors might influence the spread of an epidemic in a real-world scenario?
5. How can public health interventions impact the exponential growth rate of an epidemic?

**Tip:** Always check units when calculating exponential growth problems to ensure they are consistent (e.g., time units per annum).

Learn how to calculate the time it takes for an infected population to grow from 3 to 800 people in an epidemic with a given exponential growth rate. Includes step-by-step solution using the exponential growth formula. Suitable for advanced high school and college-level students.

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