SOLUTION: Suppose that the longest day of the year is 15 hours on June 21; the shortest day of 9 hours is on December 21; with 12 hours on March 21 and Sept 21. Write a sine equation for the

Algebra ->  Trigonometry-basics -> SOLUTION: Suppose that the longest day of the year is 15 hours on June 21; the shortest day of 9 hours is on December 21; with 12 hours on March 21 and Sept 21. Write a sine equation for the      Log On


   

Question 1103943: Suppose that the longest day of the year is 15 hours on June 21; the shortest day of 9 hours is on December 21; with 12 hours on March 21 and Sept 21. Write a sine equation for the number of daylight hours as a function of the day of the year
Answer by greenestamps(13200) About Me  (Show Source):
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We are looking for a function of the form

y=A%2Asin%28B%28x-C%29%29%2BD

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.

So our function is of the form

y=3%2Asin%28B%28x-C%29%29%2B12

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.

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.

And finally our function is

y=3%2Asin%28%282pi%2F365%29%28x-81%29%29%2B12

Here is a graph:

graph%28400%2C200%2C-30%2C400%2C-4%2C16%2C3%2Asin%28%282pi%2F365%29%28x-81%29%29%2B12%29

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