The population density of a certain microorganism in a controlled environment, D(t)D(t)D(t) in hundreds per unit area, is modelled by the function
D(t)=10ln(t+2)−t2+5,t>−2 D(t) = 10\ln(t + 2) - t^2 + 5, \quad t > -2 D(t)=10ln(t+2)−t2+5,t>−2where t t\,t is the time in hours after the start of an experiment.
Show that a time t t\,t exists in the interval [−1.3,−1.2][-1.3, -1.2][−1.3,−1.2] where the population density is zero.
The population density also returns to zero at a positive time TTT. To find the value of TTT, the iterative formula
tn+1=10ln(tn+2)+5,with t1=4 t_{n+1} = \sqrt{10\ln(t_n + 2) + 5}, \quad \text{with } t_1 = 4 tn+1=10ln(tn+2)+5,with t1=4is used.
(i) Find the value of t2 t_2\,t2 to 4 decimal places. (ii) By continuing the iteration, determine the value of T T\,T correct to 4 decimal places.
The population density reaches a maximum at a time tmaxt_{max}tmax.
Using calculus and showing each stage of your working, find the exact value of tmaxt_{max}tmax.
