Answer:
(a) 137 °F
(b) 2.0 hr
Step-by-step explanation:
Given a turkey is removed from the oven at 185 °F to a room temperature of 78 °F and cools to 150 °F after 30 minutes, you want to know the temperature after 45 minutes, and the time to cool to 100 °F.
Newton's cooling equation
The temperature described by Newton's Law of Cooling can be written as the equation ...
T(t) = (initial temperature difference) × (cooling factor)^(t/(cooling period)) + (final temperature)
Here, the initial temperature difference is 185 -78 = 107 °F. The cooling factor is (150 -78)/(185 -78) = 72/107 in a period of 30 minutes.
Thus, we can write the temperature equation as ...
T(t) = 107×(72/107)^(t/30) . . . . . . . where t is time in minutes
(a) 45 minutes
When t=45, the temperature is ...
T(45) = 107×(72/107)^(45/30) +78 ≈ 137 . . . . . degrees F
After 45 minutes, the temperature is 137 °F.
(b) 100 °F
When T = 100 °F, the time is ...
100 = 107×(72/107)^(t/30) +78
22/107 = (72/107)^(t/30)
30×log(22/107)/log(72/107) = t ≈ 119.783 . . . . minutes
t = 119.783/60 = 1.996 . . . . hours
The turkey cools to 100 °F after 2.0 hours.