Hi Marilyn,

The average of 9.6 and 14.4 is 12. 14.4 = 12 + 2.4 and 9.6 = 12 - 2.4 so the number of hours of daylight cycles between 12 - 2.4 and 12 + 2.4. Since you are assuming that the periodic cycle is a sine function the equation can de written

h(d) = 12 + 2.4 sin(f(d))

where d is the day (d = 1 on Jan 1, d = 2 on Jan 2 etc.), h is the number of hours of daylight and f(d) is some unknown expression involving d. The maximum value of the sine function is 1 and the minimum value is -1 so h(d) will cycle between 12 + 2.4 and 12 - 2.4.

You know that h(80) = 12 so when d = 80 the value of the sine function is zero. Hence the expression f(d) might be d - 80 so that when d = 80, sin(d - 80) = sin(80 - 80) = sin(0) = 0. But this can't be correct! The period of the sine function is 360 degrees and hence sin(d - 80) would cycle after 360 days and you want it to cycle once a year which is 365 days. Thus you need a scale factor k to shift the cycle from 360 units to 365 units. That is you need f(d) = sin[k(d - 80)] for some number k. Can you see a scale factor k that will work?

Harley