# Newton's law of cooling.

• Mar 11th 2010, 04:29 PM
jkeo789
Newton's law of cooling.
Hi Everyone, can someone please help me with this? It has to do with Newton's Law of cooling.

A fast food resturant wants a container that can quickly cool coffee from 200 degrees to 130 degrees and keep the liquid between 110 and 130 degrees for as long as possible. Here are the choices:
a. CK company has a containter that reduces the temp of a liquid from 200 degrees to 100 degrees in 30 mins by maintaining a constant temp of 70.

b. Temp company has a container that reduces the temp of a liquid from 200 to 110 degrees in 25 mins by maintaining a constant temp of 60 degrees.

Use Newton's law of cooling to find a function relating the temp of the liquid over time for each container.
• Mar 11th 2010, 07:01 PM
HallsofIvy
Quote:

Originally Posted by jkeo789
Hi Everyone, can someone please help me with this? It has to do with Newton's Law of cooling.

A fast food resturant wants a container that can quickly cool coffee from 200 degrees to 130 degrees and keep the liquid between 110 and 130 degrees for as long as possible. Here are the choices:
a. CK company has a containter that reduces the temp of a liquid from 200 degrees to 100 degrees in 30 mins by maintaining a constant temp of 70.

b. Temp company has a container that reduces the temp of a liquid from 200 to 110 degrees in 25 mins by maintaining a constant temp of 60 degrees.

Use Newton's law of cooling to find a function relating the temp of the liquid over time for each container.

Newton's law of cooling says that heat flows from a hotter body to a cooler body at a rate proportional to the difference between the temperatures of the two bodies. That is:
$\frac{dT}{dt}= T- c$
where "T" is the temperature of the coffee, "t" is the time of cooling and c is the temperature of the container.

For a) you have $\frac{dT}{dt}= k(T- 70)$. Integrating that will give a "constant of integration" as well as the "constant of proportion", k. To find both of those constants, use the fact that T(0)= 200 and T(30)= 100.

For b) you have $\frac{dT}{dt}= k(T- 60)$. Integrating that will give a "constant of integration" as well as the "constant of proportion", k. To find both of those constants, use the fact that T(0)= 200 and T(25)= 110.