We are given the rate of change of the total number of influenza cases over time, expressed as a function of time $t$:

$\frac{dI}{dt} = I'(t) = -6.26t + 142.6$

We are also told that $I(0) = 1418$, meaning that at $t = 0$, 1418 people had already contracted influenza. We need to find the function $I(t)$, which represents the total number of people infected by time $t$.

Step 1: Integrate $I'(t)$

To find $I(t)$, we integrate the given rate of change function with respect to $t$:

$I(t) = \int (-6.26t + 142.6) \, dt$

Step 2: Perform the integration

The integral of $-6.26t + 142.6$ is:

$I(t) = \int (-6.26t) \, dt + \int 142.6 \, dt$

This gives us:

$I(t) = -6.26 \cdot \frac{t^2}{2} + 142.6t + C$

Simplifying:

$I(t) = -3.13t^2 + 142.6t + C$

Step 3: Use initial condition to find $C$

We know that $I(0) = 1418$, so we substitute $t = 0$ into the equation to solve for $C$:

$I(0) = -3.13(0)^2 + 142.6(0) + C = 1418$

This simplifies to:

$C = 1418$

Step 4: Final equation for $I(t)$

Now, substituting $C = 1418$ into the equation for $I(t)$, we get:

$I(t) = -3.13t^2 + 142.6t + 1418$

Thus, the total number of people who have contracted influenza by time $t$ is given by:

$I(t) = -3.13t^2 + 142.6t + 1418$

Do you have any questions, or would you like more details on any part of the process?

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Here are 5 related questions to further your understanding:

1. How would the total number of cases change if the rate of infection were a different function of $t$?
2. What does the negative quadratic term in $I(t)$ tell you about the trend in the number of cases over time?
3. How would the solution change if the initial condition were different, say $I(0) = 2000$?
4. What is the significance of the constant term $1418$ in the context of this outbreak?
5. Can you find the maximum number of cases over the 20-week period?

Tip: When solving problems involving differential equations, always check if you are given initial conditions, as they are essential for finding the particular solution!