A body was found at 10 a.m in a warehouse where the temperature was 40 degrees Fahrenheit. The medical examiner found the temperature of the body to be 80 degrees Fahrenheit. What is the approximate time of death? Use Newtons law of cooling, with k = 0.1947
5 a.m, 9:30 a.m, 9:45 a.m. 8 a.m
That is algebra ? I call it Physics :)
T(t) = Tsurrounding + (Tstart -Tsurrounding) e^(-k t)
80 = 40 + ( about 99 - 40 ) e^(-.1947 t)
40 = 59 -e^.1947 T
0.678 = e^-.1947 t
ln .678 = -.389 = -.1947 t
t = 2 hours before 10 am
To find the approximate time of death, we can use Newton's law of cooling. This law states that the rate of temperature change of an object is proportional to the difference in temperature between the object and its surroundings.
The equation for Newton's law of cooling is:
dT/dt = -k(T - Ts)
Where:
dT/dt is the rate of temperature change
k is the cooling constant
T is the temperature of the object
Ts is the temperature of the surroundings
In this case, the initial temperature of the body (T0) is 80 degrees Fahrenheit, and the temperature of the surroundings (Ts) is 40 degrees Fahrenheit. The cooling constant (k) is given as 0.1947.
We need to find the time it takes for the temperature of the body to decrease from 80 degrees Fahrenheit to 40 degrees Fahrenheit.
Let's solve the equation:
dT/dt = -k(T - Ts)
dT/dt = -0.1947(80 - 40)
Integrating both sides, we get:
∫dT = -0.1947∫(80- 40)dt
T - T0 = -0.1947(80t - 40t)
T - 80 = -0.1947(40t)
Divide through by -0.1947:
(T - 80) / -0.1947 = 40t
Let's substitute the final temperature (T = 40) and solve for t:
(40 - 80) / -0.1947 = 40t
-40 / -0.1947 = 40t
t ≈ 205.55 minutes
Since the body was found at 10 a.m. and the time is given in minutes, we can calculate the approximate time of death by subtracting 205.55 minutes (or about 3 hours and 26 minutes) from 10 a.m.:
10 a.m. - 3 hours and 26 minutes ≈ 6:34 a.m.
Therefore, the approximate time of death is 6:34 a.m.
To determine the approximate time of death, we can use Newton's Law of Cooling, which states that the rate of change of an object's temperature is proportional to the difference between its temperature and the surrounding temperature.
The formula for Newton's Law of Cooling is: T(t) = T_s + (T_0 - T_s) * exp(-kt)
Where:
- T(t) is the temperature of the body at time t
- T_s is the surrounding temperature (40 degrees Fahrenheit in this case)
- T_0 is the initial temperature of the body (80 degrees Fahrenheit in this case)
- k is the cooling constant (0.1947 in this case)
- t is the time in hours
To find the time of death, we need to solve for t when T(t) is approximately equal to T_s (40 degrees Fahrenheit in this case).
Let's calculate the value of t:
40 = 40 + (80 - 40) * exp(-0.1947t)
Simplifying the equation, we have:
0 = 40 * exp(-0.1947t)
Now, divide both sides by 40:
1 = exp(-0.1947t)
To solve for t, we can take the natural logarithm (ln) of both sides:
ln(1) = ln(exp(-0.1947t))
0 = -0.1947t
Divide both sides by -0.1947:
t = 0 / -0.1947
t ≈ 0
From this calculation, it appears that the time of death is 0 hours, which means the body was not alive. However, this result might be due to a possible error in the given data or calculations. Please double-check the information provided.