Finding Time of Death

Nikki_gurl03

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Apr 12, 2007
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Here is the problem:

Suppose that a body with temperature 85F is discoveredat midnight, and that the ambient temperature is constant 70F. The body is removed quickly (assume instantly) to the morgue where the ambient temperature is maintained 40F. After one hour the body temperature is found to be 60F. Estimate the time of death.

I know the formula is dT/dt= -k(T-T(s))

T(inital)= 85F
T(s)= 70F
T(s2)=40F
t=1, T=60


I am confuse with the two different tempt. I dont know to use
T-70=Ce^(-kt)
or
T-40=Ce^(-kt)

PLease help
 
Nikki_gurl03 said:
I am confuse with the two different tempt.
What was the initial temperature reading? That one is "T(initial)"; the other one isn't.

Eliz.
 
See I am confuse with the surrounding temp that started at 70F
and then there is another surrounding temp of 40F.

So at time 0, i dont know which to use to find C.

One answer I got C=45 when I use 40F and C=15 when it is 70F

Do i have to use two different equations.

Then for k= ln(-10/15) for 70F and k=-ln(20/45)
 
Nikki_gurl03 said:
See I am confuse with the surrounding temp that started at 70F and then there is another surrounding temp of 40F.
You might want to check the definitions of the variables. I'm fairly certain that "T(initial)" does not refer to the ambient temperature.

Eliz.
 
See what the professor taught us is you get get T-(surround temp)=Ce^-kt

To solve for C you get T=the temp of the body when time is = 0

so I got
85-(surrounding)=Ce^(-k(0))

When I use 70 I get C=15 and when i use 40I get C=45

To solve for -k, you input time=(in this case)1 Temp=60

so,
60-(surround)=Ce^(-k(1))

from there I am stuck and which answers to use
 
Nikki_gurl03 said:
Here is the problem:

Suppose that a body with temperature 85F is discoveredat midnight, and that the ambient temperature is constant 70F. The body is removed quickly (assume instantly) to the morgue where the ambient temperature is maintained 40F. After one hour the body temperature is found to be 60F. Estimate the time of death.

I know the formula is dT/dt= -k(T-T(s))

T(inital)= 85F
T(s)= 70F
T(s2)=40F
t=1, T=60


I am confuse with the two different tempt. I dont know to use
T-70=Ce^(-kt)
or
T-40=Ce^(-kt)

PLease help
Firstly, note the temperature of a normal human body is 98.6F.

Consider the time from midnight onwards, set midnight as t=0. Then T(0) = 85 and T_s = 40. That information gives you an equation for T(t) after mightnight, with one unknown constant, the decay constant in the exponential. We also know T(1) = 60, so you can now calculate that constant.

Now consider the first part. You might now treat T(0) = 98.6, with T_s = 70, and with your decay constant from earlier solve T(t_0) = 85, for t_0, where t_0 is the time from death to midnight.
 
Unco said:
Firstly, note the temperature of a normal human body is 98.6F.

Consider the time from midnight onwards, set midnight as t=0. Then T(0) = 85 and T_s = 40. That information gives you an equation for T(t) after mightnight, with one unknown constant, the decay constant in the exponential. We also know T(1) = 60, so you can now calculate that constant.

Now consider the first part. You might now treat T(0) = 98.6, with T_s = 70, and with your decay constant from earlier solve T(t_0) = 85, for t_0, where t_0 is the time from death to midnight.

That's assuming the body had an average temperature at the time of death, though. Wouldn't it be more practical to assume he dies k hours before midnight and then set constraints?
 
If you set up one and solve like a system. Let \(\displaystyle t_{D}\) = length of time before midnight (time of death).

\(\displaystyle \L\\ln|85-70|=-k(0)+C, \;\ T(0)=85\)....[1]

\(\displaystyle \L\\ln|60-40|=-k(1)+C, \;\ T(1)=60\)...[2]

Solve [1] for C=ln(15).

Sub into [2] and \(\displaystyle \L\\k=ln(3/4)\)

\(\displaystyle \L\\ln|99-70|=t_{D}ln(3/4)+ln(15)\)

\(\displaystyle \L\\t_{D}=\frac{ln(29)-ln(15)}{ln(3/2)}=-2.29\)

About 2 hours and 18 minutes before midnight. 9:42 PM

I don't know if I strayed or not. Seems to make sense.
 
Thanks, I somewhat did that but I used the 40F as surround when t=1 because it was in the morgue and I used ln 45 instead of ln 15.

So, i know i am on the right track because i was doing the same thing like you but different numbers at different times
 
Nikki_gurl03 said:
See what the professor taught us is you get get T-(surround temp)=Ce^-kt
If you're supposed to use the air temperature, then I believe you have to use something like this:

. . . . .T(t) = T<sub>env</sub> + (T(0) - T<sub>env</sub>)e<sup>-rt</sup>

...where t is time, T is the temperature of the body, T<sub>env</sub> is the temperature of the surrounding environment, and r is some positive value.

You have an unknown start time (time of death), so count from the known time of midnight.

. . . . .T(0) = 85

. . . . .T(1) = 60

Since T<sub>env</sub> = 40 for this hour, then:

. . . . .60 = 40 + (85 - 40)e<sup>-r</sup>

. . . . .20 = (45)e<sup>-r</sup>

. . . . .4/9 = e<sup>-r</sup>

Solve for r.

I don't know if you use the same r when the environment has changed...? But if you can, then use this value to work backwards, in the first environment, to the time of death:

. . . . .T(t) = 70 + (85 - 70)e<sup>-rt</sup> = 98.6

I get a different time of death than the other solution. I could easily be wrong.

Eliz.
 
Nikki_gurl03 said:
morson said:
How are we supposed to know when he died?

Thats what you have to figure out.
Yes, but as I posted earlier, the assumption that they died with a body temperature of 37 degrees C is significant. I know I'm being picky, but things like whether Newton's Cooling laws are applicable in this situation (it wasn't specified in the question) get me thinking.
 
I think you're overlooking the word "estimate". This sort of thing always is based on assumptions. If you don't like the assumptions presented, then present other assumptions and solve it that way. It is not a matter of being "picky". It is a matter of accepting or rejecting the given or prevailing assumptions.

This is an interesting point in the legal system. Based on various sets of assumptions, various murder suspects could or could not have an alibi. Obviously, prosecutors and defense attorneys will have differing views concerning which assumptions should be used.
 
That's the thing. This sort of approach is not objective enough when dealing with things like murder and death. I understand that Newton realized that the decay constant would take into account things like mass and surface area that affect an object's temperature change, but as you said, it's a matter of accepting and rejecting assumptions. But the question has already been answered and I'm derailing the thread, so this is where I stop. :)
 
Do you know the 'book' answer?. What is it?. I arrived at 9:45 PM using 98.6 for the body temp when the death took place. I may have a flaw. Seems OK, though. Stapel arrived at a slightly different solution.
 
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