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°

WhereN,Candware constants, andtis the time in weeks, witht= 0 representing midnight on the first of January.

a. Taking the period of this function to be 50 weeks, find the value ofw.

b. Use the equation to describe, in terms ofNandC,

(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

a.w=7.2

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 whent=(arcos(-1))/w= 180/7.2 = 25

…?