Am influenza epidemic hits. large city and spreads at the rate of 12e^0.2t new cases per day, where t is the number of days since the epidemic began. The epidemic began 4 cases. find the number of cases during the first 30 days.

that would just be

∫[0,30] 12 e^(0.2t) dt = 24,146

To find the number of cases during the first 30 days, we need to calculate the total number of new cases each day and then sum them up.

Given that the rate of new cases per day is given by the equation 12e^(0.2t), where t is the number of days since the epidemic began, and the epidemic began with 4 cases, we can find the number of cases for each day by substituting the value of t into the equation.

Let's calculate the number of cases for each day within the first 30 days:

For t = 0 (the start of the epidemic), the number of cases is 4 (given).

For t = 1, substitute t = 1 into the equation: 12e^(0.2*1) = 12e^(0.2) ≈ 18.194.

Similarly, calculate the number of cases for t = 2, t = 3, and so on, until t = 30.

Finally, sum up the number of cases for each day from t = 0 to t = 30 to find the total number of cases during the first 30 days.