Murder at the Mayfair Diner - Solution
December 3, 2016 | Author: Dhruv Baid | Category: N/A
Short Description
A mathematical analysis of a fictional crime scene. Complete solution for the Mayfair Diner problem, which can be found ...
Description
Murder at the Mayfair – Investigation Results Dhruv K. Baid Raffles Institution November 27, 2014
Abstract This is a research paper investigating a hypothetical murder scene. Its purpose is to determine the time of death of the victim and hence decide which of the 3 possible suspects are to be detained for questioning.
1
Introduction
The body of Mr. Joe D. Wood was discovered in the refrigerator of the Mayfair Diner at 5:30 AM. He was murdered, by one of the following suspects: Twinkles the dancer, Slim the bookie, and Shorty the cook. The purpose of this paper is to determine the time of death and hence conclude which suspect should be detained, based on their times of entry and exit from the Mayfair Diner as stated by eyewitnesses.
2
Body
The first assumption is that Mr. Wood is murdered in the refrigerator where he was found. In this case, we can model the scenario using Newton’s Law of Cooling, which is given as: (
)
where represents the temperature of the body at time , and represents the external temperature. In this case, is the internal temperature of the refrigerator, given as:
Hence, (
)
∫
∫ |
| ( )
To determine the constants (
)
, we use the coroner’s readings, given by: (
)
(
)
(
)
(
( ) ( )
)
Upon substitution into ( ), we obtain the following equations: ( ) ( ) From ( ) , we obtain:
Substituting this into ( ) , we obtain two solutions:
OR
( Hence, we can substitute these values of
)
into ( ) to obtain the final solution ( ):
and
(
)
( )
into ( ), we obtain:
Substituting normal body temperature
( (
)
(
)
(
)
) (
)
Hence, according to this analysis, Mr. Wood was murdered hours, or approximately minutes, before : AM – that is, at approximately : AM that night.
hours
However, it is also possible that Mr. Wood was not murdered in the refrigerator but was instead moved there after being murdered outside. In this case, we model the situation using the following equation: (
)
where (
)
and ( ) is the unit step function defined as: ( ) Denoting
by
( (
{
) )
, we get:
(
)
To solve this equation, we take the Laplace Transformation of both sides, giving: * +
* +
* + * +
* +
* (
* +
)+
* +
( ) ( )
(
)
(
)
Hence, by using the Table of Laplace Transformations [1], we can see that {
}
{
}
{
}
{ (
( We can use the value of
}
{ (
) (
)
)
)
(
} (
{
}
)
)
obtained earlier to complete this equation: (
(
)
)
(
)
(
)
This is the equation relating the body temperature to the number of hours before : AM Mr. Wood was murdered ( hours) and moved into the refrigerator ( hours). We can now set , which is the normal human body temperature, to obtain a relation between and which will eventually show when Mr. Wood was murdered. Here, we make the assumption that Mr. Wood’s body temperature was just prior to his death. ( (
)
(
) (
)
(
)
)
(
) (
)
( )
Clearly,
, since Mr. Wood has to be murdered before his corpse can be moved. Then, (
)
Substituting this value into ( ), we have: (
(
)
) (
) (
)
Using this equation, we can create the following table relating the time the body was moved to the time of death, based on and : h 12 11 10 9 8 7 6 5 4 3 2
Time body moved 6 PM 7 PM 8 PM 9 PM 10 PM 11 PM 12 AM 1 AM 2 AM 3 AM 4 AM
Time of death 3:42 AM 3:17 AM 2:50 AM 2:20 AM 1:48 AM 1:13 AM 12:34 AM 11:52 PM 11:04 PM 10:12 PM 9:13 PM
If we assume that Mr. Wood was murdered before being moved, the results only make sense for as shown in the table above.
,
Now, we use the eyewitness accounts of the three suspects and their time of entry and exit from the Mayfair Diner to determine who the culprit is most likely to be. In my opinion, it should be Shorty the cook. Shorty is the only suspect who was in the restaurant at any of the possible times of murder (shown in green), and he only left after the corresponding time when Mr. Wood’s corpse was moved into the refrigerator. According to the eyewitness accounts, Shorty “took an unusually long break at : ”, and left “when the restaurant closed at : AM.” This makes two scenarios plausible: Mr. Wood was murdered at : PM and his corpse was moved at AM, or he was murdered at : PM and his corpse was moved at AM (assuming Shorty could move the body and escape in negligible time). Apart from this analysis, it is also worth noting that algor mortis is not perfectly explained by Newton’s Law of Cooling. A more accurate estimate could be given by the Glaister Equation. We could now derive a similar equation by using linear approximation using the equation (
)
and the initial condition ( ) The equation of the tangent passing through the point ( |
(
( ( Substituting
) is given by: )
)( ) )
( )
into ( ), we have: ( (
) )
which is similar in form to the actual Glaister Equation.
3
Conclusion
In conclusion, we have determined, using various methods of solving differential equations, the time of Mr. Wood’s death, as well as the time at which he was moved into the refrigerator.
4
Discussion
The analysis presented here was a simplification of the actual methods used in forensic investigations. Moreover, this report does not handle the motive behind the murder, which may provide further insight into this case. However, as a preliminary report, these results should suffice.
5
References
[1] Table of Laplace Transforms. Retrieved from: .
6
Appendix
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