# Researchers find a creature from an alien planet. Its body temperature is varying sinusoidally with time. 32 minutes after they start timing, it reaches a high of 124 degrees fahrenheit. 26 minutes after that it reaches its next low, 102 degrees fahrenheit. Write an equation expressing the temperature in terms of the number of minutes since they started timing. What was the temperature when they first started timing? 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`

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