Mathematics in Criminal Justice

            “There’s a dead body inside the hotel room!!! I’m really scared… Please arrive on time.” 

The police did arrive, but not quite on time. By the time they arrived, people had already gathered outside Hotel Chandani with a stressed look on their faces. The police entered the room and saw a man stained with blood on his head, and on his back — they immediately declared him dead seeing no signs of life whatsoever. “Definitely not a gunshot” - a police officer insisted as if he was a hundred percent sure. Immediately after, another police officer reached the dead body and measured his temperature while the other officers looked around if the murderer left some clue.

The police stayed in the hotel for another hour and a half: some closely inspecting the crime scene, some talking to the owner (who found it difficult to even utter a word), and some asking the gathered mass to return to their places. Before leaving the hotel, the same officer who took the body temperature reading took another temperature reading.

 A few hours later, that day, the officer confidently announced - “The murder took place at around 12:35 PM”. How could it be possible for the officer to make that claim with such confidence? Well, the officer was a forensic cop.

 Let’s try and figure out how the officer arrived at that time. Before starting the calculation, let us go into the specifics of that hour-and-half stay.

The police arrived at 2:00 pm. The first temperature reading, taken at 2:05 pm was 31°C and the next reading, taken 90 minutes after the first, read 28℃. The officer also made a crucial observation that the hotel room was air-conditioned at 25°C for the last 48 hours.

 Assuming the man was perfectly normal before the murder, i.e. a body temperature of about 37°C, it is obvious that the temperature would drop with time. But do we know something about how the temperature drops? Does it drop steadily, or in other words, does each 1°C drop in temperature require the same amount of time at a constant surrounding temperature? Does it drop slowly at the beginning and faster with time? Or does it drop quickly at the start and slows down with time? Do we know of a law of any kind that talks about the rate of drop in the temperature? Fortunately, YES!!! It’s the ‘Newton’s Law of Cooling’. 

Newton's law of cooling states that the rate of change in temperature of a body is proportional to the difference in temperature between the body and its surrounding. Before we get to its mathematical form, let us try to make sense out of this statement. What it suggests is that between two bodies that are hotter than the surrounding, the one that has a greater difference in temperature with the surrounding cools down faster than the other. 

Mathematically, when a body at temperature ‘T’ is kept in a surrounding with temperature ‘T0 ’, the rate of change of temperature at any time ‘t’ is given by 

                                      dT/dt = -k(T - T₀)

where k is a positive constant.

This is a first-order differential equation that has a general solution 

                                    T - T₀= Ce^(-kt)

 In simple words, a differential equation is an equation that relates functions with their derivatives. Since only the first derivative of T(t) appears in the equation and there are no higher-order derivatives involved, we call that a ‘first-order differential equation’. It is clear that when we put t=0 into the general solution, C = T(0) - T0 .

 In our context, T0 is the surrounding temperature of 25℃ and it stayed the same for the last 48 hours, thanks to the air conditioning in the room. So, let’s rewrite the equation as 

                                  T - 25= Ce^(-kt) 

Let us consider t=0 when the temperature reading of 31℃ was first taken. Applying this initial condition, we get

                               T - 25= 6e^(-kt)

 90 minutes later, the body temperature dropped to 28℃. Substituting this into our equation gives 

                   28 - 25= 6e^(-90k) ⇒k = 0.007702

 Now, we need to figure out the time of the murder, i.e. how many minutes before which the body temperature would be 37C. And keep in mind, since it’s the duration before the time t=0 (which we set for our convenience), we need to be prepared for a negative value of t.

                   37 - 25= 6e^(-0.007702t)⇒t = -90 min

 By this result, now we know how the officer could tell the time of the murder. One aspect of the mystery has been dealt with.

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This was just one of the applications of Mathematics in criminology; other fields of Mathematics used in criminology include trigonometry, probability, graph theory, linear algebra, etc. For example, trigonometry helps in analyzing the blood splatter and deduces the likely scenario that caused such splatter, which in turn helps in crime scene reconstruction when combined with other evidence.

- Bimarsh Adhikari