Bacteria A bacteria culture starts with 1,000 bacteria and doubles in size...

Bacteria A bacteria culture starts with 1,000 bacteria and doubles in size every 3 hours. Find an exponential model for the size of the culture as a function of time t in hours and use the model to predict how many bacteria there will be after

2 days. HINT [See Example 4.]

Example 4:

In the early stages of the AIDS epidemic during the 1980s, the number of cases in the United States was increasing by about 50% every 6 months. By the start of 1983, there were approximately 1,600 AIDS cases in the United States.*

a. Assuming an exponential growth model, find a function that predicts the number of people infected t years after the start of 1983.

b. Use the model to estimate the number of people infected by October 1, 1986, and also by the end of that year.

Solution

Alternatively, if we wish to use the method of Example 2, we need two data points. We are given one point: (0, 1,600). Because y increased by 50% every 6 months, 6 months later it reached 1,600 + 800 = 2,400 (t = 0.5). This information gives a second point: (0.5, 2,400). We can now apply the method in Example 2 to find the model above.

b. October 1, 1986, corresponds to t = 3.75 (because October 1 is 9 months, or 9/12 = 0.75 of a year after January 1). Substituting this value of t in the model gives

By the end of 1986, the model predicts that

(The actual number of cases was around 41,700.)