# find temp

• Dec 3rd 2009, 07:00 PM
Morgan82
find temp
A pan of water (46 degrees) was put into a fridge. Ten min. later, the temp was 39; 10 min. later, the temp was 33.
Use newtons law of cooling to find the temp of the fridge..
I know the equation.

Do I need to find the rate, or how do I do this?
• Dec 3rd 2009, 08:16 PM
nehme007
By Newton's law of cooling you have that the temperature of the water is given by:
$T(t) = T_f + (46 - T_f) e^{-rt}$
where $T_f$ is the temperature of the fridge and r is some unknown rate constant. You are told T(10) = 39 and T(20) = 33, so:
$T(10) = 39 = T_f + (46 - T_f) e^{-10r}$
$T(20) = 33 = T_f + (46 - T_f) e^{-20r}$
The first equation gives you:
$r = -1/10 ln(\frac{39-T_f}{46 - T_f})$
and the second gives you:
$r = -1/20 ln (\frac{33-T_f}{46 - T_f})$.
Equating these two expressions for r and multiplying by -20 yields:
$2ln(\frac{39-T_f}{46 - T_f}) = ln (\frac{33-T_f}{46 - T_f})$.
The 2 can move inside the log as a power, then you can get rid of the logs and solve. You should get $T_f = -3$.