An ecologist is monitoring two distinct populations of insects.
The number of insects, NNN, in the first population is modelled by the equation
N=Aekt,t≥0 N = A e^{kt}, \quad t \ge 0 N=Aekt,t≥0where A A\,A and k k\,k are positive constants and t t\,t is the time in days from the start of the observation.
Given that:
Find the exact value of A A\,A and the value of k k\,k to 4 significant figures.
The number of insects, NNN, in the second population is modelled by the equation
N=80000e−0.4t,t≥0 N = 80000 e^{-0.4t}, \quad t \ge 0 N=80000e−0.4t,t≥0where t t\,t is the time in days from the start of the observation.
Find the rate of decrease of insects in this second population exactly 4 days from the start. Give your answer to 3 significant figures.
When t=Tt = Tt=T, the number of insects in the two populations was the same.
Find the value of TTT, giving your answer to 3 significant figures.
(Solutions relying entirely on calculator technology are not acceptable.)
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.
