A population of bacteria in a laboratory culture is being monitored. The number of bacteria, BBB, in the culture, t t\,t hours after the initial observation, is modelled by the equation
B=400ekt7+ekt B = \frac{400e^{kt}}{7 + e^{kt}} B=7+ekt400ektwhere k k\,k is a constant.
Find the number of bacteria in the culture at the start of the study.
Given that there are 160 bacteria in the culture after 5 hours,
show that k=15ln(143)\displaystyle k = \frac{1}{5}\ln\left(\frac{14}{3}\right)k=51ln(314).
Given also that, when t=Tt = Tt=T, the number of bacteria is increasing at a rate of 25 per hour,
find the possible values of TTT, giving your answers to one decimal place.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
