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#26 2011-02-02 06:39:17
- Reuel
- Member
- Registered: 2010-11-28
- Posts: 178
Re: Newton's Law of Cooling - Differential Equation
It's been fun.
Here is another heating/cooling problem. Following is my solution. How does it sound?
"The police discover the body of a murder victim at 12pm. and find the temperature of the body to be 94.6 F. The body temperature of the victim is then 93.4 F one hour later. The temperature of the room is 70 F. When was the victim murdered?"
Yes, gruesome. Here is my solution:
Initial Conditions: T(0) = 94.6 F, T(1) = 93.4 F, and the change in body temperature over one hour is 1.2 F.
Solving for C:
And solving for k,
Assuming the body was 98.6 F when the murder occurred...
Solving for t gives the time of death to be 15 minutes before the initial temperature was taken. This sounds reasonable to me because cooling is exponential.
How's that sound?
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#27 2011-02-02 06:58:04
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Newton's Law of Cooling - Differential Equation
Hi;
You use the second condition at T(1) = 93.4 to determine k.
So you should change that 1.2 to 93.4 and re solve for k.
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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#28 2011-02-02 07:15:34
- Reuel
- Member
- Registered: 2010-11-28
- Posts: 178
Re: Newton's Law of Cooling - Differential Equation
That is what I did at first and I got some crazy answer like negative 27 hours. Here it is worked out:
Now I am getting t = -0.052575
Which is nonsense. Unless it means the event occurred minutes beforehand.
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#29 2011-02-02 07:26:56
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Newton's Law of Cooling - Differential Equation
Hi;
I am getting -3.01 hours remember the time of death has to be before T(0). That seems to make sense with the temperature drop.
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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#30 2011-02-02 07:30:40
- Reuel
- Member
- Registered: 2010-11-28
- Posts: 178
Re: Newton's Law of Cooling - Differential Equation
How did you get that? I am getting different values every time I calculate it.
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#31 2011-02-02 07:47:10
- 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(1) to solve for k the constant of proportionality.
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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#32 2011-02-02 08:40:23
- Reuel
- Member
- Registered: 2010-11-28
- Posts: 178
Re: Newton's Law of Cooling - Differential Equation
I see my mistake. Thanks...
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#33 2011-02-02 08:45:46
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Newton's Law of Cooling - Differential Equation
Okay, your welcome.Just fix the time of death at 3 hours ago.
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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