Newton's Law of Cooling

T(t)=T0+(T1-T0)e^-kt

The police discover the body of a murder victim. Critical to solving the crime is determining when the murder was committed. The coroner arrives at the murder scene at 12:00pm. She immediately takes the temperature of the body and finds it to be 94.6 degrees F. She then takes the temperature 1 hour later and finds it to be 93.4 degrees F. The temperature of the room is 70 degrees F. When was the murder committed?

Let t=0 at time of death.

We have two temperatures, 94.6 at t=t1, and 93.4 at t=t1+1.
Also, normal body temperature is 98.6, so T1=98.6.
T0=ambient temperature = 70.
Substitute in given formula:
T(t)=T0+(T1-T0)e^-kt

94.6=70+(98.6-70)e^(k*t1)...(1)
93.4=70+(98.6-70)e^(k*(t1+1))...(2)

We try to solve for t1 and k.
Subtract (2) from (1):
Rearrange (1) and (2)
e^(k*t1)=0.86014 ...(1a)
e^(k*(t1+1))=0.81818 ...(2a)

Divide (2a) by (1a)
e^(k*(t1+1)-k*t1)=0.81818/0.86014
e^k=0.95122
take natural log on both sides,
k=-0.0500

Substitute k in (1a):
e^(-0.05t1)=0.86014
take natural log on both sides
-0.05t1 = -0.15066
t1=-0.15066/(-0.05)
=3.01

So the victim was dead 3 hours before the coroner arrived, or at 9 a.m.

What is the answer if the coroner arrives at the scene at 1 a.m. and immediately takes the temperature of the body and is 94.6F. She then takes it 1 hour later and it is 93.1F. The outside temperature is 60F

To determine when the murder was committed using Newton's Law of Cooling, we can set up an equation using the given information.

The equation for Newton's Law of Cooling is:
T(t) = T0 + (T1 - T0)e^(-kt)

Given information:
T0 = temperature of the room = 70 degrees F
T(0) = initial temperature of the body = 94.6 degrees F
T(1) = temperature of the body after 1 hour = 93.4 degrees F

We want to find the value of t, which represents the time when the murder was committed.

Substituting the given values into the equation, we have:
94.6 = 70 + (93.4 - 70)e^(-k*0)
24.6 = 23.4e^(-k*0)

Simplifying further:
24.6 = 23.4e^0
24.6 = 23.4

Since this equation is not true, we made a mistake in our calculation. Let's try again.

Using the correct equation for Newton's Law of Cooling:
T(t) = T0 + (T1 - T0)e^(-kt)

We know:
T(0) = 94.6
T(1) = 93.4
T0 = 70

Plugging in these values, we have:
93.4 = 70 + (T1 - 70)e^(-k*1)

Simplifying:
93.4 - 70 = (T1 - 70)e^(-k)
23.4 = (T1 - 70)e^(-k)

Now, we need another piece of information to solve for k and find the time of the murder.

To determine the time of death using Newton's Law of Cooling, we can use the formula:

T(t) = T0 + (T1 - T0) * e^(-kt)

where:
- T(t) represents the temperature of the body at time t
- T0 is the initial temperature of the body (at the time of death)
- T1 is the temperature of the room
- k is the cooling constant

Given information:
- T0 = 94.6 degrees F (temperature of the body at time 0, which is 12:00 pm)
- T1 = 70 degrees F (temperature of the room)
- T(1 hour) = 93.4 degrees F (temperature of the body 1 hour later)

We can now substitute the given values into the equation:

93.4 = 94.6 + (70 - 94.6) * e^(-k * 1)

Simplifying the equation further:

93.4 = 94.6 - 24.6 * e^(-k)

Rearranging the equation:

e^(-k) = (94.6 - 93.4) / 24.6

e^(-k) = 1.2 / 24.6

Now, we take the natural logarithm of both sides of the equation to solve for k:

ln(e^(-k)) = ln(1.2 / 24.6)

-k = ln(1.2 / 24.6)

k ≈ -ln(1.2 / 24.6)

Using a calculator or computer program, we find:

k ≈ 0.06415

Now that we have the value of k, we can determine the time of death by plugging it into the equation:

T(t) = 94.6 + (70 - 94.6) * e^(-0.06415 * t)

To find t, the time of death, we solve the equation for t when T(t) = 70:

70 = 94.6 + (70 - 94.6) * e^(-0.06415 * t)

Subtracting 94.6 from both sides of the equation:

-24.6 = -24.6 * e^(-0.06415 * t)

Dividing both sides by -24.6:

1 = e^(-0.06415 * t)

Now, we take the natural logarithm of both sides in order to isolate t:

ln(1) = ln(e^(-0.06415 * t))

0 = -0.06415 * t

Simplifying, we find:

t ≈ 0

Based on the calculations, it appears that the time of death was approximately 0 hours, or at the time when the coroner arrived at 12:00 pm.