Chapter 4, Problem 59RE

Spreading of a Disease Jack and Diane live in a small town of 50 people. Unfortunately, both Jack and Diane have a cold. Those who come in contact with someone who has this cold will themselves catch the cold. The data that follow on the next page represent the number of people in the small town who have caught the cold after $t$ days.

$\begin{array}{cc}\text{Days, t }& \text{Number of people with Cold, C}\\ 0& 2\\ 1& 4\\ 2& 8\\ 3& 14\\ 4& 22\\ 5& 30\\ 6& 37\\ 7& 42\\ 8& 44\end{array}$

Using a graphing utility, draw a scatter plot of the data. Comment on the type of relation that appears to exist between the number of days that have passed and the number of people with a cold.

Using a graphing utility, build a logistic model from the data.

Graph the function found in part (b) on the scatter plot.

According to the function found in part (b), what is the maximum number of people who will catch the cold?

In reality, what is the maximum number of people who could catch the cold?

Sometime between the second and third day, $10$ people in the town had a cold. According to the model found in part (b), when did 10 people have a cold?

How long will it take for $46$ people to catch the cold?