Exercise 6

Suppose that $b(t)$ measures the number of bacteria living in a colony in a Petri dish, where $b$ is measured in thousands and $t$ is measured in days. One day, you measure that there are 6,000 bacteria and the per capita growth rate is 3. A few days later, you measure that there are 9,000 bacteria and the per capita growth rate is 2.

1. Assume that the per capita growth rate $\frac{db/dt}{b}$ is a linear function of $b$. Use the measurements to find this function and write a logistic equation to describe $\frac{db}{dt}$.

2. What is the carrying capacity for the bacteria?

3. At what population is the number of bacteria increasing most rapidly?

4. If there are initially 1,000 bacteria, how long will it take to reach 80% of the carrying capacity?