# Thread: Newton's law of cooling

1. ## Newton's law of cooling

The rate at which a substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of the air is 300K and the substance cools from 370K to 340K in 25 minutes, find when the temperature will be 310K.

2. Originally Posted by cazimi
The rate at which a substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of the air is 300K and the substance cools from 370K to 340K in 25 minutes, find when the temperature will be 310K.
$
\frac{d(T)}{dt} = k(T - 300)$

$
\frac{d(T)}{T-300} = k dt$

Integrate both sides

$ln|T-300| =kt + h$

At t = 0 ,T = 370K

Hence h =ln|70|

ln|370-340| - ln|70| = 25k

ln|30/70| = 25k
k = (ln|3/7| )/25

c =25 ln|70| / ln|3/7|

Put this in the equation and T = 310 to get the value of t

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Another thing the sign of k has come out to be negative thus we can see that the temperature is decreasing, (something you can observe by reading the question)