I need to know how to write the function with the steps needed to solve.

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  f(t) = 17.5cos(π(t-7)/6) +59

Step-by-step explanation:

When modeling earth data that has a daily or annual cyclic variation, the simplest sort of model is a sine or cosine function of some sort. Generally, its period will be 1 day or 1 year (or about 1/2 day, if you're modeling tides).

To write the trig function, you need to know ...

These values will go into the general form ...

  f(t) = A·cos(2π(t -C)/B) +D

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In this particular form, the cosine function has a value of 1 when t=C, so we can choose C to be the value of t where f(t) is a maximum. In your data table, the maximum temperature is in July, where you have said t=7. So, C=7.

The period is 12 months, so B=12.

The amplitude (A) is half the difference between the minimum and maximum:

  A = (76.5 -41.5)/2 = 17.5

The offset (D) is the average of the maximum and minimum:

  D = (76.5 +41.5)/2 = 59

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Using these values in the above form, we have ...

  f(t) = 17.5cos(2π(t-7)/12) +59

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Additional comments

You can see in the attachment that we have matched the maximum and minimum values exactly. A more sophisticated model is needed to adjust the function to match more of the data points.

Or, you may get a slightly better match if you use a sine regression model (least-squares best fit). That sort of modeling is best done using a graphing calculator or spreadsheet. The details are beyond the scope of this answer.

i need the options

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