During weeks in a certain flu season, the rate at which the number of cases of influenza per 100,000 people in a country changed could be approximated by I'(t)=3.298e^0.1044t where I is the total number of people per 100,000 who have contracted influenza and t is time measured in weeks. Estimate I(t), the total number per 100,000 who have contracted influenza by time t. Assume that I(0)=0

I have no idea how to do this. Please help. Thank you!

I(0) = 3.298e^0.1044*0,

I(0) = 3.298e^0, e^0 = 1.
I(0) = 3.298 * 1,
I(0) = 3.298 cases per 100,000.

To estimate the total number per 100,000 who have contracted influenza at a given time, we can integrate the given rate equation I'(t) with respect to time to find the cumulative number of cases over time.

The given rate equation is I'(t) = 3.298e^(0.1044t). To find I(t), we need to integrate I'(t) with respect to t.

The formula for integration is:

∫ [ f(t) ] dt = F(t) + C

Where ∫ represents the integral sign, f(t) represents the function we want to integrate, F(t) is the antiderivative of f(t), and C is the constant of integration.

In this case, we have:

∫ [ 3.298e^(0.1044t) ] dt = I(t) + C

To integrate the function 3.298e^(0.1044t), we recognize that it is the derivative of the exponential function e^(0.1044t) with respect to t. Hence, the antiderivative is:

∫ [ e^(0.1044t) ] dt = (1/0.1044)e^(0.1044t) + C

Now, we can substitute this back into the original equation:

I(t) = (1/0.1044)e^(0.1044t) + C

To find the specific value of C, we can use the initial condition I(0) = 0. Substituting t = 0 and I(t) = 0 into the equation, we get:

0 = (1/0.1044)e^(0.1044 * 0) + C

0 = (1/0.1044) + C

C = -1/0.1044

Now we can substitute C back into the previous equation:

I(t) = (1/0.1044)e^(0.1044t) - 1/0.1044

This equation represents the total number per 100,000 who have contracted influenza by time t in the given flu season, assuming I(0) = 0.

To estimate a specific value for I(t), you would need to substitute the desired time value t into the equation.