Answer :
To determine the equation that models the temperature of the object as a function of time, we will use Newton's Law of Cooling.
According to Newton's Law of Cooling, the temperature of an object changes at a rate proportional to the difference between its own temperature and the ambient temperature. The formula given is an exponential decay function:
[tex]T(t) = T_r + (T_0 - T_r) e^{-kt}[/tex]
where:
- [tex]T(t)[/tex] is the temperature of the object at time [tex]t[/tex].
- [tex]T_0[/tex] is the initial temperature of the object (150°F in this case).
- [tex]T_r[/tex] is the room temperature (60°F in this case).
- [tex]k[/tex] is the rate of cooling constant.
- [tex]t[/tex] is the time in minutes.
Here, the function is given in the form:
[tex]T(t) = 90 e^{-kt} + 60[/tex]
The initial temperature difference is [tex]T_0 - T_r = 150 - 60 = 90[/tex], which is consistent with the equation provided.
Now, we will use the information given to find [tex]k[/tex]. After 2 minutes, the temperature of the object has decreased to 130°F. Using this information:
[tex]130 = 90 e^{-2k} + 60[/tex]
Subtract 60 from both sides:
[tex]70 = 90 e^{-2k}[/tex]
Divide both sides by 90:
[tex]\frac{70}{90} = e^{-2k}[/tex]
Simplify the fraction:
[tex]\frac{7}{9} = e^{-2k}[/tex]
To solve for [tex]k[/tex], take the natural logarithm of both sides:
[tex]\ln\left(\frac{7}{9}\right) = -2k[/tex]
Solve for [tex]k[/tex]:
[tex]k = -\frac{1}{2} \ln\left(\frac{7}{9}\right)[/tex]
Calculate the value of [tex]k[/tex]:
[tex]k \approx -\frac{1}{2} (-0.2513) \approx 0.126[/tex]
Thus, the constant [tex]k[/tex], rounded to the nearest thousandth, is approximately [tex]0.126[/tex].
Therefore, the function that models the temperature of the object as a function of time is:
[tex]T(t) = 90 e^{-0.126t} + 60[/tex]
