After sitting out of a refrigerator for a while, a turkey at room temperature (72, degrees72
∘
F) is placed into an oven. The oven temperature is 315, degrees315
∘
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, equals, T, start subscript, a, end subscript, plus, left bracket, T, start subscript, 0, end subscript, minus, T, start subscript, a, end subscript, right bracket, e, start superscript, minus, k, t, end superscript
T=T
a

+(T
0

−T
a

)e
−kt

T, start subscript, a, end subscript, equalsT
a

= the temperature surrounding the object
T, start subscript, 0, end subscript, equalsT
0

= the initial temperature of the object
t, equalst= the time in hours
T, equalsT= the temperature of the object after tt hours
k, equalsk= decay constant

The turkey reaches the temperature of 136, degrees136
∘
F after 2 hours. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 6 hours.

You want the decay constant k and the temperature after 6 hours of a 72 °F turkey put into a 315 °F oven, given that it has a temperature of 136 °F after 2 hours.

Newton's law of cooling

The given formula is ...

[tex]T(t)=T_a+(T_0-T_a)e^{-kt}[/tex]

We are given the values ...

[tex]T_a=315\\T_0=72\\T(2)=136[/tex]

Decay constant

Using these values, we can write the equation for the decay constant.

[tex]136 = 315 +(72 -315)e^{-k\cdot2}\\\\-\!179=-243e^{-2k}\\\\\ln\left(\dfrac{179}{243}\right)=-2k\\\\k=-\dfrac{1}{2}\ln\left(\dfrac{179}{243}\right)\\\\\boxed{k\approx0.153}[/tex]

6 hours

The temperature after 6 hours is then ...

[tex]T(6)=315-243e^{-0.153\cdot6}\\\\\boxed{T(6)\approx218}[/tex]