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.)
