Question

The population of a caribou herd varies sinusoidally with a period of one year. The population is at a minimum at the beginning of the year when the population is about 3000 animals. The population is at a maximum of 8000 animals after 6 months. How many caribou are there expected to be at the end of 9 months? Round to the nearest whole number.

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Answer 1

The range from the minimum to the maximum is 5000,
so a = 2500
let's use a sine curve, so far we have
count = 2500 sin(kt), where t is the number of years
period = 1, so 2π/k = 1
k = 2π
So far, count = 2500 sin(2πt)

This would give us a min at t = -1/4 but we want our min to be at t = 0
so let's do a phase shift of π/2 to the right
count = 2500 sin 2π(t - 1/4)

almost there ....
When t = 0 we get a minimum of -2500
but we want that min to be 3000, so raise it up 5500

final: count = 2500 sin 2π(t - 1/4) + 5500

check: when time is 6 months, t =1/2
count = 2500 sin 2π(10-1/4) + 5500 = 8000

we want the count when time is 9 months or t = 3/4
count = 2500 sin 2π(3/4 - 1/4) + 5500 = 5500

www.wolframalpha.com/input/?i=graph+y++%3D+2500+sin%28+2%CF%80%28t+-+1%2F4%29%29+%2B+5500

This equation is not unique, we could have used a cosine curve, or used
a different phase shift.