# the sun rises at 9:17 on december 21 and at 04:35 on june 22. Period=365days write a sinusoidal equation for sun rise time to the day of the year?

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### 1 Answer

You need to remember the general form of sinusoidal equation such that:

`f(t) = a*sin(bt+c) + d`

You need to determine a,b,c,d such that:you need

`a = (9:17 - 4:35)/2` (you need to divide by 2 since there are two terms involved in differerence)

You need to convert 9:17 in minutes such that:

`9:17 = 9*60 + 17 = 557 ` minutes

`4:35 = 4*60 + 35 = 275` minutes

`a = (557 - 275)/2`

`a = 282/2 =gt a = 141`

You may find d by averaging themaximum and minimum minutes such that:

`d = (557 + 275)/2`

`d = 416`

The problem provides the information that the period is of 365 days, hence t = 365 such that:

`b*365 = 2pi =gt b = 2pi/365`

The problem provides the information that at 22 June, the sine function reaches its minimum (the value of sine is -1). The sine function reaches its minimum when completes 3/4 from the cycle of 365 days.

You need to find how many days passed by till the day of 22 June such that:

`31 + 28 + 31 + 30 + 31 + 22 = 173 `

Hence, 22 June is the 173 rd day of the year, hence `c = 173+365/4` .

`c = 264.25`

**Hence, evaluating the sinusoidal function that models the sun rise time yields `f(t) = 141sin(2pi/365*t + 264.25) + 416` .**