# In 2001, Windsor, Ontario received its maximum amount of sunlight, 15.28 hrs, on June 21, http://postimage.org/image/ady4eo2z5/ Applications

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### 1 Answer

We are given that the maximum sunlight occurs on June 21 (day 172) and is 15.28 hours. Also the minimum hours of sunlight is 9.08 on Dec. 21 (day 155)

The function of daylight hours is periodic and can be modeled with a sinusoid.

A sinusoid is `y=asin[b(x-h)]+k` where a gives the amplitude, the period can be found from b (`p=(2pi)/b` ), h is the phase shift (horizontal translation), and k is a vertical translation and gives themidline.

(1) The amplititude is `(15.28-9.08)/2=3.1` a=3.1

(2) The period is 365 days, so `b=(2pi)/p=(2pi)/365`

(3) The maximum of a sine wave beginning at the origin occurs at 1/4 of the cycle , or on day 91.25, so the phase shift is 80.75 days to the right, so h=80.75

(4) The midline is `y=(15.28+9.08)/2=12.18` so k=12.18

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An equation modeling the amount of sunlight is :

`y=3.1sin[(2pi)/365(x-80.75)]+12.18`

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The graph: