a. To create an exponential model for new cases in terms of days, we can use the formula: y = a * b ^ x, where y is the number of new cases, x is the number of days since the first observation, and a and b are constants that we need to determine. Using the two data points given, we can set up a system of equations:
235 = a * b ^ 0
395 = a * b ^ 10
Solving for a and b, we get:
a = 235
b = (395/235)^(1/10) = 1.067
Therefore, the exponential model for new cases in Milwaukee is:
y = 235 * 1.067 ^ x
b. To find the number of new cases on Oct. 31, 2020, we need to plug in x = 19 (since Oct. 31 is 19 days after Oct. 12) into the model:
y = 235 * 1.067 ^ 19 = 1018.5
Therefore, based on the exponential model, we would expect around 1019 new cases on Oct. 31, 2020.
c. The actual number of new cases on Oct. 31, 2020, was 1043. This is higher than the predicted value of 1019, but not by a huge margin. Overall, the model seems to fit the data reasonably well, especially considering that there are many factors that can affect the number of new cases in a given area, and that the model is based on only two data points. However, it is worth noting that the exponential model assumes that the growth rate of new cases remains constant over time, which may not be a realistic assumption in the long run.