# Newton's Law of Cooling problem

• May 17th 2007, 07:49 AM
zocchihedron
Newton's Law of Cooling problem
Alright, I'm having issues with this problem. You all remember when that lady spilled coffee on herself and then sued mcdonalds for saying it was too hot? Well i have a problem that's based on that. You can view the problem here. Tommy Ratliff - Writing in Calculus (sorry i'm too lazy to type it all out, lol)

Anyways, in essence there are two periods of time. THe first one he's holding the coffee in the cup holder of his car (T(s)=72 degrees F) and then the second one he's holding it in his hands (T(s)=92.3 degrees F). Anyhow, what i wanna do is find an inequality that if he held the cup holder and then his hands for greater than this amt of time then the final temp will be above 140 etc. etc.

So assuming that the entire t will be 10 min (600s) and that the T(0)=160 i set it up like so:
k= ln.875/-270
Final T (FT) = 140 <-- that's what we want
and the FT for after he's put it in the cup holder for so long will = the T(0) for when he's holding it in his hands. Hence

T=(160 -72) e^*((600-t)*(ln(.875)/270))+72
T=T(0)
140-92.3=(T(0) - 92.3)e^(t*ln(.875)/270) ~ substitute the T(0) for
the expression above

But that didn't work... and it's driving me nuts.... I'm sorry, i know i didn't explain it very well, but if anyone actually gets what i mean and knows what i did wrong or how to do it, then id GREATLY appreciate it.
• May 17th 2007, 10:53 AM
CaptainBlack
Quote:

Originally Posted by zocchihedron
Alright, I'm having issues with this problem. You all remember when that lady spilled coffee on herself and then sued mcdonalds for saying it was too hot? Well i have a problem that's based on that. You can view the problem here. Tommy Ratliff - Writing in Calculus (sorry i'm too lazy to type it all out, lol)

Anyways, in essence there are two periods of time. THe first one he's holding the coffee in the cup holder of his car (T(s)=72 degrees F) and then the second one he's holding it in his hands (T(s)=92.3 degrees F). Anyhow, what i wanna do is find an inequality that if he held the cup holder and then his hands for greater than this amt of time then the final temp will be above 140 etc. etc.

So assuming that the entire t will be 10 min (600s) and that the T(0)=160 i set it up like so:
k= ln.875/-270
Final T (FT) = 140 <-- that's what we want
and the FT for after he's put it in the cup holder for so long will = the T(0) for when he's holding it in his hands. Hence

T=(160 -72) e^*((600-t)*(ln(.875)/270))+72
T=T(0)
140-92.3=(T(0) - 92.3)e^(t*ln(.875)/270) ~ substitute the T(0) for
the expression above

But that didn't work... and it's driving me nuts.... I'm sorry, i know i didn't explain it very well, but if anyone actually gets what i mean and knows what i did wrong or how to do it, then id GREATLY appreciate it.

If you do this properly you will find that the coffee is cooling for 4.5 minutes
at an ambient temprature of 72f from 160 to 149 f. Then if the ambient
temprature is 92f the coffee will cool to 140f in an additional 2 minutes (approx)
assuming the same rate constant.

However this is invalid because you will not have the same rate constant
with the cup being held as with it standing. This is because you no longer
have free flow of air around the cup. You cannot conclude anything without
doing the experiment to determine the appropriate rate constant for a held
cup.

RonL