To find $I(t)$, the total number of people infected by time $t$, we need to integrate the given rate of change function $I'(t) = -6.56t + 141.6$.

Step-by-Step Solution:

1. Given: $I'(t) = -6.56t + 141.6$ and $I(0) = 1419$.

2. Integrate $I'(t)$ to find $I(t)$:

   The integral of $I'(t)$ will give us $I(t)$: $I(t) = \int (-6.56t + 141.6) \, dt$

3. Compute the integral: $I(t) = \int (-6.56t) \, dt + \int 141.6 \, dt$ $I(t) = -6.56 \cdot \frac{t^2}{2} + 141.6t + C$ Simplify: $I(t) = -3.28t^2 + 141.6t + C$

4. Determine the constant $C$ using $I(0) = 1419$:

   Substitute $t = 0$ and $I(0) = 1419$: $I(0) = -3.28(0)^2 + 141.6(0) + C = 1419$ So, $C = 1419$.

5. Final equation for $I(t)$: $I(t) = -3.28t^2 + 141.6t + 1419$

This is the formula for the total number of people who have contracted influenza by time $t$ in weeks.

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Do you want more details on any step or have any questions?

Related Questions:

1. What is the estimated number of people infected at $t = 10$ weeks?
2. When will the number of infected people start decreasing?
3. How many people will have contracted influenza at the end of 16 weeks?
4. What is the maximum number of people infected at any given time?
5. Can you explain how the rate of infection changes over time?

Tip:

Always verify your solution by differentiating $I(t)$ to see if you get back $I'(t)$. This confirms the integration is correct.