Respuesta :
Answer: Mr. Ratchett died approximately at 2AM.
Step-by-step explanation: Newton's Law of Cooling shows the cooling rate between a body and the environment, by stating that the rate is proportional to the difference in temperatures between them.
Mathematically, it is represented as:
[tex]\frac{dQ}{dt}=\alpha.A.(T_{s}-T)[/tex]
This differential equation solved, gives the solution:
[tex]T(t)=T_{s}+(T_{0}-T_{s})e^{-kt}[/tex]
where
[tex]T_{s}[/tex] is temperature of the environment
[tex]T_{0}[/tex] is the initial temperature of the body
k is a parameter dependent of heat transfer coefficient, heat capacity and area of the body
t is time required to change the temperature
For the murder on the Orient Express, first determine parameter k.
In 1 hour, Mr. Ratchett's body decrease 1°:
[tex]27=22+(28-22)e^{-1k}[/tex]
[tex]5=6e^{-k}[/tex]
[tex]e^{-k}=\frac{5}{6}[/tex]
[tex]ln(e^{-k})=ln(0.834)[/tex]
k = 0.1823
Using the parameter, find estimate time:
[tex]28=22+(37-22)e^{-0.1823t}[/tex]
[tex]6=15e^{-0.1823t}[/tex]
[tex]ln(e^{-0.1823t})=ln(0.4)[/tex]
0.1823t = 0.9163
[tex]t=\frac{0.9163}{0.1823}[/tex]
t ≈ 5 hours
The body was found at 7AM. So, it took approximately 5 hours to cool down to 28° at that time. Therefore, Mr. Ratchett died at 2AM.
