Question 1103943
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We are looking for a function of the form<br>
{{{y=A*sin(B(x-C))+D}}}<br>
The number of hours of daylight varies between 9 and 15.  If we represent the number of daylight hours with a sine function, then the center line is 12 and the amplitude is 3.  The A in the formula is the amplitude; the D is the center line.<br>
So our function is of the form<br>
{{{y=3*sin(B(x-C))+12}}}<br>
Since we want a sine function, we want the phase shift to be such that the function value is at its central value 12 and increasing on March 21, which is day 81 in the year.  So, since our function is to be in terms of the day of the year, C in our function is 81.<br>
And finally, the B in our formula is (2pi) divided by the period of the function, which is 365 days.  So B is (2pi)/365.<br>
And finally our function is<br>
{{{y=3*sin((2pi/365)(x-81))+12}}}<br>
Here is a graph:<br>
{{{graph(400,200,-30,400,-4,16,3*sin((2pi/365)(x-81))+12)}}}<br>
The values given by the function at the specified dates are the following:
March 21 (day 81): 12
June 21 (day 173): 14.9998
September 21 (day 265): 11.922
December 21 (day 356): 9.0007