Let's start with part a, finding the maximum and minimum.
Answer a: Max temp is -5 C on day 200. Min temp is -79 C on day 500.
Sine model.
We're looking for a model
T(n) = A sin( ωn - p ) + o
where T(n) is the estimated temperature on day n, A the amplitude of the variation, omega the radial frequency of variation, p a phase shift and o an offset, the average temperature.
Averaging min and max gives -42 C; that's the offset, what we get when the sine is zero. o = -42 C.
That's pretty close to the value on day 0 so we'll just use a sine function without any phase shift, p = 0.
Subtracting min from max gives a total span of 74 C degrees, so a sinusoidal amplitude that's half that, A = 37 C.
From max to min is a half period, or about a half period since we're only seeing every 100 days. That's 300 days, so a full cycle about every 600 days.
We want the argument to sine to go from 0 to 2pi as we complete a single period, 600 days, So when n=600 this term must be 2π, so it's of the form
ωn = 2 π n / 600.
Putting it all together we get our model:
Answer b: T(n) = 37 sin(2 π n / 600) - 42
We already estimated the period at 600 days. That's a full year.
Answer c: 600 days
We can see a full period is closer to 700 days (compare day 500 and day 1200, and also day 600 and day 1300) and the phase shift might need adjusting, but I think we pretty much did as they asked so I'll stop here.
