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 18◦ Celsius

. (b) If the temperature of the victim’s body is now 25◦, 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?

Answer 1

Solytion: a) Newtons law 아 cool, dt kt given, T-18C K .1941 T18 C31-18) T(t) 18t 19e-0.747t model b) Given Tit): 25% o 1947t 2518t e o.1947t 0.1947t0.118528330 t 5.128GhourS t 5hours 7roioutes 43second< T(t)- 18+ (36.3-18)e947t 25-18 - 18-3e 0.1947t 0.147t -0.9699090908 t 4.358 bour s It 4hours 56minutes 9 seconds Oißßerence osenved is Ilminutes 34seconds

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