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#1 2011-02-02 01:41:49

Reuel
Member
Registered: 2010-11-28
Posts: 178

Newton's Law of Cooling - Differential Equation

A heating a cooling differential equation word problem:

"Blood plasma is stored at 40 degrees F. Before it can be used, it must be at 90 degrees F. When the plasma is placed in an oven at 120 degrees F, it takes 45 minutes for the plasma to warm to 90 degrees F. Assume Newton's Law of cooling applies. How long will it take the plasma to warm to 90 degrees F if the oven temperature is set at 100 degrees F?"


Looking up Newton's Law of Cooling, I believe a good starting place is to state the equation as

where T is the temperature at time t and that change is directly proportional to the difference between the surrounding temperature of the chamber and that of the object.

Listing the initial values...

(for a 120F oven)


And so, what the problem is asking for is for us to find the time it will take to warm the plasma all the way up to 90 F if the oven is only 100. Just working off of the definition of Newton's Law of Cooling, here is a first attempt at setting up the problem:



That isn't right but I am not sure how to specify two different temperatures for two different objects, the oven and the plasma, t 0 and time t.

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#2 2011-02-02 01:50:16

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Wait, the temperature of the oven, in this model, is constant. Could I just write

?

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#3 2011-02-02 02:01:45

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Hi Reule;

I have it as -k in my notes?!


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#4 2011-02-02 02:05:46

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Hi bobbmy.

I am just going by what my teacher said.

"The rate of cooling of an object is directly proportional to the difference in temperature of the surroundings and the temperature of the object."

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#5 2011-02-02 02:08:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Why not solve the DE you were given and then just plug in.

Where R is the temperature of the surrounding environment.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#6 2011-02-02 02:08:59

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

What DE was I given? Other than the cooling law on the right side which I just based on my notes.

Last edited by Reuel (2011-02-02 02:10:32)

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#7 2011-02-02 02:15:07

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Hi;

I mean the one you found:


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#8 2011-02-02 02:17:27

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Because yours

and mine

have the order of the temperature of the object and the temperature of the surrounding area reversed. Which one is the proper order?

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#9 2011-02-02 02:22:37

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Hi Reule;

I do not think it will matter. Give me a minute to complete my work and let us see if we can agree on a solution.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#10 2011-02-02 02:57:42

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Hi

Solve the DE.

Solve for c using the initial conditions.

Use the next value at T(45) to solve for k the constant of proportionality.

Where R is the surrounding temperature.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2011-02-02 03:58:01

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

I think solving for k was throwing me off... I wasn't sure what conditions to use for it. A lot of my trouble with these problems is knowing how to use the information given. I suppose that's true for most people.

Here is my work doing it the other way. I am not sure if our answers are the same or not.

Solving for C with t = 0,

Solving for k by following your example,

So the function would then be

If I have done my math correctly, that is.

I don't know. What do you think?

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#12 2011-02-02 04:07:04

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Try to answer the next part of the question with your formula, I get around 95 minutes.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2011-02-02 04:16:16

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

t = 2? That isn't right. I must have messed something up.

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#14 2011-02-02 04:20:25

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Yes, that is not correct. Your k is not right. If 120 took 45 minutes at 100 it has to take longer, a lot longer.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2011-02-02 04:24:05

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

So, apparently, it does matter whatever order the difference is put in? Or did I make some little error? It's a good thing to know. smile

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#16 2011-02-02 04:27:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Hi Reuel;

Yes, I think so now.
Check out this page here and then see if you can do the example they have.

http://www.math.wpi.edu/Course_Material … node5.html

It is what I use for these cooling problems.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2011-02-02 04:30:07

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Awesome! Thanks! That is going into my notebook for sure.

Let me the problem that way. Hang on.

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#18 2011-02-02 04:32:31

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Okay, if you do not get her answer than we can look at it. Is that okay with you?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2011-02-02 04:42:15

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Yes.

In post #10 how did you get 8/3? I keep coming up with 3/8.

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#20 2011-02-02 04:45:44

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

They are the same mine has a - sign in front.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#21 2011-02-02 04:51:52

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Less than two years ago I hardly knew much more than how to solve 3x = 6 for x. In that time I have gotten as far as I have and somehow, someway, in all that time I never realized a negative sign made equal the reciprocal of the inside of a logarithm. How did that ever escape me? And for so long?

You learn new stuff every day.

Finishing up the problem now...

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#22 2011-02-02 05:01:02

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

I got 95.4038 minutes.

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#23 2011-02-02 05:03:31

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

That I is correct, I think.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#24 2011-02-02 05:56:43

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Newton's Law of Cooling - Differential Equation

Horray. Here is my work:




Solving first for C with the initial condition T(0) = 45 which is where T_s is 120 and where the plasma is 40 F...


Therefore, the equation is now

Solving for the proportionality constant k the initial conditions are used again:


So the new function solved for C and k is

The new function is in terms of T_s and t and so to calculate how long it will take in time (t) for the plasma to warm to 90 degrees (T(t)) the equation can be solved for t:


And by that t comes out to be approximately 95 minutes.


Thanks for the help. I now see how solving for c and then k using the initial conditions allows for the establishment of a function with which to calculate new values for time.

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#25 2011-02-02 06:04:04

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Newton's Law of Cooling - Differential Equation

Hi Reuel;

Glad to help! Thanks for working with me on them.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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