# Math Help - Newton's law of cooling.

1. ## 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.

2. 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.