Question

Solve this really good question

Answer 1

When a person dies, their body begins to cool. The temperature of the body at any time after death is governed by Newton’s Law of Cooling, which applies to any cooling object : \[D=a\times 10^{kt}\]
Where
\(D=\text{Is the temperature difference between the cooling object and its surroundings}\)
\(a\text{ and }k=\text{Constants that need to be calculated}\)
\(t=\text{The time since the object started to cool}\)

The police arrive at the scene of a murder at 8 a.m. On arrival the temperature of the body and its surroundings are measured as 34° and 17° respectively. This was taken to be the time when t is zero.

At 9 a.m., the body temperature was measured as 33° and room temperature was still 17°.
Estimate the time of death given the body temperature of a living person is about 36.9°.

Answer 2

@Directrix. Please help

Answer 3

@Zarkon

Answer 4

Good one isn't it :)

Answer 5

Any idea, @Directrix @Zarkon?

Answer 6

I believe I know how to start
Equation: \(D=a*10^{kt}\)
We know at the beginning that at 8a.m., the body is 34 degrees, and the outside is 17. \(t=0\) at this point. Let us sub in the values:
\[D=a*10^{kt}\]\[(34-17)=a*10^{k*0}\]\[17=a*10^{k*0}\]\[17=a*10^0\]\[17=a*1\]\[17=a\]

Answer 7

It states that the equation is \[D=a*10^{ft}\]

Answer 8

Not \(f\), \(k\)

Answer 9

@Directrix?

Answer 10

@ParthKohli, do you know?

Answer 11

@iGreen, please help

Answer 12

\[D = a \times 10^{kt}\]I believe \(a\) is the temperature at the time of death, which is \(36.9\) C, and \(k\) decides the rate. Let's use the information we're given.\[34 - 17=36.9 \times 10^{kt}\]\[33 - 17 = 36.9 \times 10^{k(t+1)}\]

Answer 13

It says they are just constants

Answer 14

They are.

Answer 15

How do we go from there?

Answer 16

It says \(t\) is zero for the first one

Answer 17

We don't know if \(t\) is in minutes or hours

Answer 18

Whoops, \(a\) is not the temperature at the time of death, but the temperature difference.

Answer 19

Isn't that \(D\)?

Answer 20

I believe \(a\) is the initial temperature, as that fits the idea of the growth/decay formula of \(y=ka^x\)

Answer 21

Yeah, \(a\) is \(D\) at \(t=0\).

Answer 22

So what should we do?

Answer 23

\[17 = a \times 10^{kt}\]\[16 = a \times 10^{k(t+1)}\]\[\Rightarrow 17/16 = 10^{kt - kt - k}\]\[\Rightarrow 17/16 = 10^{-k}\]So you know the value of \(k\).

Answer 24

What is it?

Answer 25

\(log_{10}(\frac{17}{16})=-k\)?

Answer 26

\[10^k = 16/17\]\[k = \log(16/17)\]

Answer 27

You put negative sign before

Answer 28

\[10^{-k} = \frac{17}{16}\]\[\Rightarrow \dfrac{1}{10^{-k}} = \dfrac{1}{\frac{17}{16}}\]\[\Rightarrow 10^k = \frac{16}{17}\]

Answer 29

OH, K

Answer 30

So, k=-0.026?

Answer 31

So you can write your equation as\[D = a \times \left(\frac{16}{17}\right)^t\]

Answer 32

Hey, thats what I got before ;)

Answer 33

No, it would be:
\[D=a\times10^{t(\frac{16}{17})}\]
Wouldn't it?

Answer 34

\[10^k = \frac{16}{17}\]Raise both sides to power \(t.\)

Answer 35

Since we subbed in \(k\) into:
\[D=a\times10^{kt}\]

Answer 36

OH

Answer 37

Man, you are \[\color{purple}{\large{GOOD}}\]

Answer 38

So, now what?

Answer 39

Find \(a\)?

Answer 40

My example before found 17, not sure if it is right

Answer 41

\[17 = a \times \left(\frac{16}{17}\right)^{t}\]\[16 = a \times \left(\frac{16}{17}\right)^{t+1}\]Thinking...

Answer 42

Can we estimate the room temperature at the time of death to be 17 degrees?

Answer 43

I think thats the only way?

Answer 44

In that case, the value of \(a = 36.9 - 17\).

Answer 45

How?

Answer 46

Since \(a\) is the temperature difference at \(t=0\).

Answer 47

t = 0 is the time of death, by the way.

So the temperature of the body is 36.9 and the room temperature, as we have supposed, is 17.

Answer 48

But should D be 36.9-17, since that is the difference in temperature, according to the question?

Answer 49

At \(t=0\), \(a = D_{t=0}\). Check it using the equation \(D = a \times 10^{kt}\).

Answer 50

So we can find the value of \(a.\)

Answer 51

So \(a=19.9\)?

Answer 52

Yes.

Answer 53

So the equation is:
\[D=19.9\times(\frac{16}{17})^t\]
Right?

Answer 54

Yeah.

Answer 55

So now what?

Answer 56

To find the time?

Answer 57

Of the death of the person?

Answer 58

I guess you guys are tired. If you reply, thanks :) I appreciate it. Going to take a rest for a bit. Have a medal

Answer 59

\[17 = 19.9 \times (16/17)^t\]