# Newton's law of cooling

• Aug 5th 2012, 07:11 AM
leonm92
Newton's law of cooling
Newton's law of cooling leads to a mathematical model of the cooling of a hot cup of coffee. The temperature http://place32.placementtester.com:8...ohaejhkjcb.gif of the coffee at any time http://place32.placementtester.com:8...mchminjkgh.gif is given by the solution of the following differential equation

http://place32.placementtester.com:8...ceggkefjjc.gif

Find the general solution of the differential equation

Find the particular solution of the differential equation if initially, the temperature of the coffee is 83 degrees.

Find the temperature of the coffee when t = 4.3

How do I do this?
• Aug 5th 2012, 07:48 AM
HallsofIvy
Re: Newton's law of cooling
$\frac{dT}{dt}+ 0.2T- 5= 0$ is the same as $\frac{dT}{dt}= 5- 0.2T$.
Changing to differential form $\frac{dT}{5- 0.2T}= dt$. Can you integrate those?
• Aug 5th 2012, 07:49 AM
skeeter
Re: Newton's law of cooling
Quote:

Originally Posted by leonm92
Newton's law of cooling leads to a mathematical model of the cooling of a hot cup of coffee. The temperature http://place32.placementtester.com:8...ohaejhkjcb.gif of the coffee at any time http://place32.placementtester.com:8...mchminjkgh.gif is given by the solution of the following differential equation

http://place32.placementtester.com:8...ceggkefjjc.gif

Find the general solution of the differential equation

Find the particular solution of the differential equation if initially, the temperature of the coffee is 83 degrees.

Find the temperature of the coffee when t = 4.3

How do I do this?

separation of variables ...

$\frac{dT}{dt} = 5 - 0.2T$

$\frac{dT}{dt} = 0.2(25 - T)$

$\int \frac{dT}{25-T} = \int 0.2 \, dt$

finish it and get temperature $T$ as a function of time $t$ ...
• Aug 6th 2012, 02:25 AM
leonm92
Re: Newton's law of cooling
to be honest, I haven't got a clue about integrating
• Aug 6th 2012, 05:38 AM
skeeter
Re: Newton's law of cooling
Quote:

Originally Posted by leonm92
to be honest, I haven't got a clue about integrating

then it's obvious you are ill prepared to tackle this problem ... here is a link for a place to start

Indefinite Integrals Part 1 | Khan Academy Calculus Lecture
• Aug 6th 2012, 06:39 AM
HallsofIvy
Re: Newton's law of cooling
Quote:

Originally Posted by leonm92
to be honest, I haven't got a clue about integrating

It is not a good idea to take differential equations until after you have learned calculus.