The population of a culture of bacteria has a growth rate given by 200 p'(t) = (t

The population of a culture of bacteria has a growth rate given by 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 The growth rate decreases in time, reflecting competition for space and food.)

a. Using the population model with r = 2, what is the increase in the population over the time interval 0 ≤ t ≤ 4?

b. Using the population model with r = 3, what is the increase in the population over the time interval 0 ≤ t ≤ 6?

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.

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.

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?

Transcribed Image Text:

200 p'(t) = (t + 1)" JP'(s) ds.