Question

after sitting in a refrigerator for a while, a turkey at a temperature of 38°f is placed on the counter and slowly warms closer to room temperature (72°f). newtons 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 room, as given by the formula below: $t = t_a+(t_0 - t_a)e^{-kt}$ $t_a=$ the temperature surrounding the object $t_0=$ the initial temperature of the object $t =$ the time in minutes $t=$ the temperature of the object after $t$ minutes $k =$ decay constant the turkey reaches the temperature of 45°f after 35 minutes. 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 120 minutes. enter only the final temperature into the input box.

Explanation:

Step1: Identify the given values

$T_a = 72$, $T_0=38$, $t = 35$, $T = 45$

Step2: Substitute values into the formula

$45=72+(38 - 72)e^{-35k}$

Step3: Simplify the equation

$45=72-34e^{-35k}$
$34e^{-35k}=72 - 45$
$34e^{-35k}=27$
$e^{-35k}=\frac{27}{34}$

Step4: Take the natural - logarithm of both sides

$-35k=\ln(\frac{27}{34})$
$k=-\frac{\ln(\frac{27}{34})}{35}$
$k\approx0.007$

Step5: Find the temperature after 120 minutes

Now that $k\approx0.007$, $T_a = 72$, $T_0 = 38$, $t = 120$
$T=72+(38 - 72)e^{-0.007\times120}$
$T=72-34e^{-0.84}$
$T=72-34\times0.432$
$T=72 - 14.7$
$T\approx57$

Answer:

57