Chapter 5.5, Problem 87E

Population models The population of a culture of bacteria has a growth rate given by $p^{\prime }(t)=\frac{200}{{(t+1)}^r}$ bacteria per hour, for t ≥ 0, where r > 1 is a real number. In Chapter 6 it is shown that the increase in the population over the time interval [0, t] is given by ${\displaystyle {\int }_0^t{p}^{\prime }(s)ds}$. (Note that the growth rate decreases in time, reflecting competition for space and food.)

1. a. Using the population model with r = 2, what is the increase in the population over the time interval 0 ≤ t ≤ 4?
2. b. Using the population model with r = 3, what is the increase in the population over the time interval 0 ≤ t ≤ 6?
3. c. Let ΔP be the increase in the population over a fixed time interval [0, T]. For fixed T, does ΔP increase or decrease with the parameter r? Explain.
4. d. A lab technician measures an increase in the population of 350 bacteria over the 10-hr period [0, 10]. Estimate the value of r that best fits this data point.
5. e. Looking ahead: Use the population model in part (b) to find the increase in population over the time interval [0, T], for any T > 0. If the culture is allowed to grow indefinitely (T → ∞), does the bacteria population increase without bound? Or does it approach a finite limit?