MATH PEOPPE PLEASE HELP
After sitting out of a refrigerator for a while, a turkey at room temperature (72°F) is
placed into an oven. The oven temperature is 305°F. Newton's Law of Heating
explains that the temperature of the turkey will increase proportionally to the
difference between the temperature of the turkey and the temperature of the oven, as
given by the formula below:
T = T. + (T. – T.)e-kt
To = the temperature surrounding the object
To = the initial temperature of the object
t the time in hours
T = the temperature of the object after t hours
k = decay constant
The turkey reaches the temperature of 123°F after 2 hours. Using this information,
find the value of k, to the nearest thousandth. Use the resulting equation to
determine the Fahrenheit temperature of the turkey, to the nearest degree, after 4.5
hours.
Enter only the final temperature into the input box.

Newton's Law of Heating results in the exponential function you show in your problem statement. We can use the given starting and "in-progress" temperatures and times to solve for the value of the factor "k" in the exponential function.

[tex]T(t)=T_\infty+(T_0-T_\infty)e^{-kt}\\\\T(2.5)=123=305+(72-305)e^{-k\cdot2.5}\qquad\text{substitute given temps and times}\\\\\dfrac{123-305}{72-305}=e^{-2.5k}=\dfrac{182}{233}\qquad\text{subtract 305, divide by $(72-305)$}\\\\k=\dfrac{-1}{2.5}\ln\dfrac{182}{233}\approx0.0988127\approx0.099\qquad\text{rounded to 3 places}[/tex]

Then the temperature after 4.5 hours is ...

T(4.5) = 305 -233e^(-0.099·4.5) ≈ 156 . . . . °F

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Additional comment

It will take about 6.3 hours for the turkey temperature to reach 180 °F.