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embizze | High School Teacher | (Level 1) Educator Emeritus

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(1) Determine an equation that can model the amount of sunlight:

The model will be a sinusoid. There are many different correct versions, so here is one:

We know the amount of sunlight hits a maximum on day 172 (June 21), so we can use a `cos` model, shifted right 172 units.

The period we assume to be 365 days.

The amplitude is `("max"-"min")/2=(15.28-9.08)/2=3.1`

The midline is at `y="max"-"amplitude"==>y=12.18`

The equation for a cosine model is `y=AcosB(x-h)+k` where A is the amplitude, B is `(2pi)/"period"` ,h is the horizontal translation, and k the vertical translation.

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An equation to model the amount of sunlight is :

`y=3.1cos[.0172(x-172)]+12.18`

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The graph:

You could ,of course, use a sine model.

(2) When can they expect 13.5 hours of sunlight?

About days 107 and 237 (From the graph and a table).

`13.5=3.1cos[.0172(x-172)]+12.18`

`.4258=cos(.0172(x-172))`

`.0172x-2.9584=cos^(-1)(.4258)`

`.0172x=4.089`

`x=237`  But `cos^(-1)(.4289)` can also be in the other quadrant

`.0172x-2.9584=5.165 ==> x=472.3` and `472-365=107`

 

Sources:

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