# Math Help - Exponential function about determing the time of death.

1. ## Exponential function about determing the time of death.

Using Newton's Law of Cooling, I have to predict the time of death of a person.

H-R = Be^-kt is the relationship between the temperature of the body (H) and the constant room temperature (R), t is the time in hours since the body expired and k and B are the constants. Using that equation I have to determine the time of death.

When the murder is committed, the body, originally at 37'C, cools according to Newton's Law of Cooling, if the body is found at 9:30am the temperature of the body is 35'C and that temperature of the body is 30'C at 4:00pm. Assuming that the temperature of the surrounding air is a constant 20'C.

I know how to solve simpler exponential models, but I'm confused how to do this one.

2. Originally Posted by eriiin
Using Newton's Law of Cooling, I have to predict the time of death of a person.

H-R = Be^-kt is the relationship between the temperature of the body (H) and the constant room temperature (R), t is the time in hours since the body expired and k and B are the constants. Using that equation I have to determine the time of death.

When the murder is committed, the body, originally at 37'C, cools according to Newton's Law of Cooling, if the body is found at 9:30am the temperature of the body is 35'C and that temperature of the body is 30'C at 4:00pm. Assuming that the temperature of the surrounding air is a constant 20'C.

I know how to solve simpler exponential models, but I'm confused how to do this one.
At $t_0=0$ we have:

$H-R=17=Be^0=B$

So $B=17$.

At $t=9.5$ (we are working in decimal hours here):

$H-R=15=17e^{-k(9.5-t_0)}$

At $t=16.0$ (4 pm is 16.00hrs):

$H-R=10=17e^{-k(16.0-t_0)}$

Now take logs and solve the resulting equations for $t_0$ and $k$

CB

3. I'm a little confused, what does the o in to mean? And why is it 17d^-k(9.5-t)?

4. Originally Posted by eriiin
I'm a little confused, what does the o in to mean? And why is it 17d^-k(9.5-t)?
$t_0$ denotes the time of death

The second question I don't understand, $9.5$ hours is 09:30 so $9.5-t_0$ is the time since death at 09:30 in hours

CB

5. Sorry I should have been more specific, I just don't understand why it's 9.5 - t, like why is that we subtract it?

6. Originally Posted by eriiin
Sorry I should have been more specific, I just don't understand why it's 9.5 - t, like why is that we subtract it?
It is the time from death at 09:30

CB