To solve this problem, we model the growth of the disease using exponential growth.

Step 1: Define the exponential growth equation

The number of people infected $P(t)$ after $t$ weeks is given by: $P(t) = P_0 \cdot 2^t$ where:

• $P_0$ is the initial number of infected people,
• $2^t$ accounts for the doubling every week.

Here, $P_0 = 3$, and we want to find $t$ such that $P(t) = 5000$.

Step 2: Substitute values into the equation

$5000 = 3 \cdot 2^t$

Step 3: Solve for $t$

1. Divide both sides by 3: $\frac{5000}{3} = 2^t$ $1666.67 = 2^t$

2. Take the base-2 logarithm of both sides (or use natural logarithm $\ln$ and convert): $t = \log_2(1666.67)$

Using the conversion formula for logarithms: $\log_2(x) = \frac{\ln(x)}{\ln(2)}$ $t = \frac{\ln(1666.67)}{\ln(2)}$

Step 4: Compute the value of $t$

$\ln(1666.67) \approx 7.418 \quad \text{and} \quad \ln(2) \approx 0.693$ $t = \frac{7.418}{0.693} \approx 10.71$

Step 5: Interpret the result

Since $t$ represents time in weeks, it will take approximately 11 weeks for the entire station of 5000 people to become infected, since the disease spreads exponentially.

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Related Questions

1. What would happen if the disease spreads every 3 days instead of weekly?
2. How many people are infected after 6 weeks?
3. What if the initial number of infected individuals was 10 instead of 3?
4. How does the doubling rate impact the time to infect the entire station?
5. How long would it take to infect half the station (2500 people)?

Tip:

When solving exponential growth problems, remember to use logarithms to simplify calculations and solve for time!