Q)
At 12:00 an apple pie is removed from the oven and placed on a table to cool. The temperature
of the room is 24 degrees C. At 12:20 the temperature of the pie is 36 degrees C and is decreasing at a rate of 2degrees/min. What was the temperature of the pie when it was brought out of the oven?
A)
Okay, So I believe I have done most of the steps and I am sure I am near the end of this problem, but I am not sure how to proceed from this certain step that I arrived at.
So, we know that at $$t=20 min$$,
we get
$$36=24+(T_{initial}-24)e^{20k}$$
Since we know what dT/dt is we can plug it into the formula and get $$k = 1/6$$
Now, I plug that into equation and get
$$36=24+(T_{initial}-24)e^{20\cdot1/6}$$
I am stuck at this part of how to isolate and find Tinitial, can someone show me the algbra steps taken to solve for Tinitial which is the Temp of the pie when brought out of oven.
The answer is 12e^(10/3) + 24, but I have no idea how to get this answer.
Best Answer
You have calculated the value of k wrong. Since the temperature is decreasing the value of $k$ will be $$k=-\frac 16$$ $$36=24+(T_{initial}-24)e^{-20\over 6}$$ $$12=(T_{initial}-24)e^{-10\over 3}$$ $$12e^{10\over 3}=T_{initial}-24$$ $$T_{initial}=12e^{10\over 3}+24$$