In: Other
Question 1. Estimating the time of a victim’s death
during homicide investigations is a complex problem that cannot be
solved by analysising simple equations or functions of one
variable. However, many mathematical texts examine time of death
estimation based around analysis of Newton’s Law of Cooling. Such
analysis is based on implicit simplifying assumptions that: the
only dependent variable of interest in determining the time of
death is the victim’s body temperature, T(t); the victim’s baseline
body temperature when alive, T0, is known; and the air temperature
of the victim’s surroundings, Ts, is constant. Here we will examine
such a problem.
(a) Assume that immediately following death, a victim’s body begins
to cool from a standard healthy body temperature of 37◦ Celsius.
Further, assume that experimental work has determined that the rate
constant in Newton’s Law of Cooling for a human body is
approximately k = 0.1947 when time t is measured in hours.
Determine a function derived from Newton’s Law of Cooling, T(t),
that models the temperature of a victim’s body t hours after death,
assuming that the temperature of the body’s surroundings is a
constant 15.5◦ Celsius.
(b) If the temperature of the victim’s body is now 22.2◦, how long ago was their time of death?
(c) If the victim’s body temperature at death had instead been 36.3◦ Celsius (within the range of normal body temperatures for a healthy adult), what time of death would be estimated via a Newton’s Law of Cooling model? By what duration does this estimate differ to the time that you determined in part (b)?
(d) In reality, how might the modelling assumptions made to
address this problem be violated?