The equation of the temperature T as a function of time t is of the form:

`T=asin(b(t-c))+d`

where a is the amplitude, `(2pi)/b` is the period, c is the horizontal shift or phase and d is the vertical shift.

Calculate a

`a=(124-102)/2=11`

The difference between the time at the high...

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The equation of the temperature T as a function of time t is of the form:

`T=asin(b(t-c))+d`

where a is the amplitude, `(2pi)/b` is the period, c is the horizontal shift or phase and d is the vertical shift.

Calculate a

`a=(124-102)/2=11`

The difference between the time at the high and the time at the low is equal to half the period, then:

`pi/b=26min`

`b=pi/26`

And the period is `26x2=52min`

To calculate the horizontal shift, determine the t intercept before the high.

`t=32-26/2=19`

therefore c=19

The vertical shift is equal to the average of the high and low.

`d=(124+102)/2=113`

Therefore the equation would be:

`T=22sin(pi/26(t-19))+113` (1)

To compute the temperature, substitute 0 for t in (1)

`T=11sin((-19pi)/26)+113=104.8`o

Thus the temperature when they started timing was 104.8 degrees.

The function can also be expressed as a cosine with a phase shift to the left. The minimum before the maximum at t=32 is at t=32=26=6, therefore the next maximum is at t=20 and the cosine function is:

`T=11cos(pi/26(x+20))+113`