Jane comes home and finds her pie half eaten on the window s

Jane comes home and finds her pie half eaten on the window sill. She wants to find out which one of her roommates did this. The temperature of the room is 72 degrees Fahrenheit. The refrigerator is 35 degrees Fahrenheit. At 10:30 pm the pie\'s internal temperature is 62 degrees Fahrenheit and half an hour later it rises to 63.5 degrees Fahrenheit.

1. When was the pie taken out of the refrigerator? Was it Abby (in the room until 7 pm) or Rachel (in the room after 7 pm)?

2. What assumptions did you have to make?

Solution

Assume that when the pie was taken out of the fridge, its temperature was 35 deg F
Now temp at 10:30 = 62
temp at 11:00 = 63.5
Assume that the room temperature is the same since the pie was taken out, and that the heat capacity of the pie does not change with change in temperature
so, dT = k(To - T)dt
Ti[-ln(To - T) ]Tf = k[tf - ti]
ln[To - Tf]/[To - Ti] = kt

k(30) = ln([72-63.5]/[72-62]) = ln(8.5/10) = -0.1625
k = -0.005417

so ln[To - Tf]/[To - Ti] = -0.005417t
ln[72 - 63.5]/[72 - 35] = -0.005417t
t = 271.5 mins = 11.00 - 4:31.5 = 6:29.5 --> Abby