The number of hours of sunlight in a day can be modeled by a sinusoidal function

Question

The number of hours of sunlight in a day can be modeled by a sinusoidal function. In the northern hemisphere, the longest day of the year occurs at the summer solstice and the shortest day occurs at the winter solstice. In 2000, these dates were June 22 (the 172nd day of the year and December 21 (the 356th day of the year), respectively. A town experiences 11.17 hours of sunlight at the summer solstice and 8.13 hours of sunlight at the winter solstice. Find a sinusoidal function y A sin (wx p) +B that fits the data, where x is the day of the year. (Note: There are 366 days in the year 2000.) 27t y 1.52 sin 183 3 9.65 1727t 27t y 11.17 sin 356 3 9.65 27t y 11.17 sin 7tx 8.13 1617t y 1.52 sin 9.65 183 366

Explanation / Answer

Given that town experiences 11.17 hrs of summer solstice

And 8.13 hrs of winter solatice

The value of sine varies from

[-1,1].

Hence the function value is of form

1.52 sin (y) +9.65

Here determine phase change is given for a period of 366 daya as mentioned.

The maximum value of solsitise occur at x = 172 days hence on substituing the value in the option 4 it satisfies for summer and winter solsitise

Hence option 4 is the required sinusoidal equation.

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