The population, P, of a certain type of bird on a remote island varies during the course of a year according to feeding, breeding, migratory seasons and predator interactions. An ornithologist doing research into bird numbers for this species attempts to model the population on the island with the annually periodic equation
P = N – Ccoswt°
Where N, C and w are constants, and t is the time in weeks, with t = 0 representing midnight on the first of January.
a. Taking the period of this function to be 50 weeks, find the value of w.
b. Use the equation to describe, in terms of N and C,
(i) the number of birds of this species on the island at the start of each year;
(ii) the maximum number of these birds, and the time of year when this occurs
My answers were
b. (i) P=N-C
These were correct
For b. (ii) I gave
P=N+C, t=25 weeks
However the answer is P=N+C for t=37.5 weeks.
How is this possible?
P=N+C when coswt°=-1, therefore when t= (arcos(-1))/w = 180/7.2 = 25