A biologist is studying the population, PPP, of a specific strain of bacteria in a petri dish. The rate of change of the population is modeled by the differential equation
dPdt=3P(4−t)8 \frac{dP}{dt} = \frac{3P(4 - t)}{8} dtdP=83P(4−t)where t≥0t \ge 0t≥0 is the time in hours since the start of the experiment. Initially, the population is 40 units.
Solve the differential equation to show that the population at time ttt is given by
P=40e316(8t−t2)for 0<t<c P = 40 e^{\frac{3}{16}(8t - t^2)} \quad \text{for } 0 < t < c P=40e163(8t−t2)for 0<t<cwhere ccc is a constant to be found that represents the time when the population first returns to its initial value.
Find the exact maximum population predicted by this model. Fully justify that your answer is a maximum.