Determine when the person was murdered?

Question

Police arrive at a murder scene at 3:30a.m and immediately record the body's temperature which was at 92F. At 5:00a.m after thoroughly inspecting and fingerprinting the area, they again took the temperature of the body which had dropped to 85F. The temperature of the crime scene has remained at a constant 70F. Determine when the person was murdered.(Assume the temperature of the body at the time of death was 98.6F)

(round to the nearest minute as needed)

Answer 1

Let T(t) = temperature of the body t hours after the temperature                        was first recorded

 

By Newton's Law of cooling, T(t) = T_s + (T₀ - T_s)e^-kt, where T_s is the temperature of the surroundings, T₀ is the temperature the first time that it is recorded, and k is a constant to be determined.  

 

So, we have T(t) = 70 + (92 - 70)e^-kt

 

                               = 70 + 22e^-kt

 

Next, we need to find k:  We are given that the temperature at       5 AM was 85°.  Since 5 AM is 1.5 hours after 3:30 AM, we get:

 

T(1.5) = 70 + 22e^-1.5k = 85

 

                         22e^-1.5k = 15

 

                             e^-1.5k = 0.681818

 

                               -1.5k = ln(0.681818)

 

                                      k ≈ 0.255

 

So, T(t) = 70 + 22e^-0.255t

 

We are trying to find the time of death.  In other words, since it is assumed that the body temperature was normal at the time of death, we need to find t so that T(t) = 98.6°.

 

70 + 22e^-0.255t = 98.6

 

         22e^-0.255t = 28.6

 

             e^-0.255t = 1.3

 

               -0.255t = ln(1.3)

 

                          t = -1.0288795 hours (the negative sign indicates                                       that the time of death was 1.0288795 hours                                           prior to the time that the temperature of the                                         body was first taken)

 

Thus, the time of death was 1.0288795 hours before 3:30 AM.

 

1.0288795 hours = 1.0288795(60) minutes ≈ 62 minutes

 

Time of death = 3:30 AM - 62 minutes = 2:28 AM

 

 

Answer 2

Hi Britteny,

 

Before I answer any problem, I set up visual clues to help me.  For example, I would write down this information for this problem as such:

 

 (Time murder took place)           X = 98.6F

            (Police first arrive) 3:30am = 92.0F

(Police finish investigation) 5:00am = 85.0F

 

To answer this problem, you must first think in differences between time and body temperatures.  For example, you need to find:

 

1. The difference between the time when police first arrive and the second time after their thorough investigation and..

 

2. The difference between the time of the murder (X)(The X is the time we need to figure out when the murdered occurred.) and the time the police first discover the body.

 

The next step is to figure out the differences between body temperature when first murdered and the body temperature when police first arrive.  You must also find the difference between the body temperature when police first arrive and the body temperature when police finish their investigation.  

 

If you set it up as ratios, with X being the time the body is murdered, you can cross multiply and find your answer.

 

If you are still puzzled, let me help you and we can set up a tutoring session.

 

Sharon T.