using Newton's Law of Cooling: At 10:30 am the medical examiner arrives at a crime scene where a dead body has been found. The temperature of the body is 80 degrees F and the thermostat for the room in which the body was found is set at 68 degrees F. One hour later the temperature of the body is 78.5 degrees F. When did death occur?
To determine when the death occurred using Newton's Law of Cooling, we need to find the time it takes for the body's temperature to decrease from the initial temperature (80 degrees F) to the room temperature (68 degrees F).
Newton's Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature.
The general form of Newton's Law of Cooling equation is:
dT/dt = -k(T - Ta)
Where:
dT/dt is the rate of change of temperature with respect to time,
k is the cooling constant,
T is the temperature of the object,
Ta is the ambient temperature.
To solve this equation, we need to calculate the cooling constant (k) first. We can do this by using the given information:
At time t = 0 (10:30 am), the body's temperature (T) is 80 degrees F, and the ambient temperature (Ta) is 68 degrees F.
Using the second piece of information, after one hour (t = 1 hour), the body's temperature (T) is 78.5 degrees F.
We can substitute these values into the equation to solve for the cooling constant (k):
dT/dt = -k(T - Ta)
(78.5 - 68) = -k(80 - 68)
10.5 = -12k
Solving for k:
k = -10.5/12
k = -0.875
Now that we have the cooling constant (k), we can determine when the death occurred by finding the time it takes for the body's temperature to decrease from 80 degrees F to 68 degrees F:
dT/dt = -k(T - Ta)
dT/dt = -(-0.875)(80 - 68)
dT/dt = 0.875 * 12
dT/dt = 10.5
From the equation, dT/dt represents the rate of change of temperature with respect to time (t).
Since we are given that the body's temperature decreases from 80 degrees F to 68 degrees F, we know that the temperature change is -12 degrees F.
dT/dt = ΔT/(Δt)
10.5 = -12/Δt
Solving for Δt (time taken for the temperature to change from 80 degrees F to 68 degrees F):
Δt = -12/10.5
Δt = -1.14 hours
To find out when death occurred, we need to subtract the time it took for the temperature change (-1.14 hours) from the time when the medical examiner arrived at the crime scene (10:30 am).
10:30 am - 1.14 hours = 9:18 am
Therefore, based on Newton's Law of Cooling, the death occurred at approximately 9:18 am.