You come into your professor's office to ask some questions shortly before 9:00 a.m. on Friday.

You find him lying on the floor of his office in a pool of chalk dust, dead.

You quickly call the police and their investigators take several measurements over the next hour, including:
1) the body temperature at 9:00 a.m. - 80 degrees.
2) the body temperature at 10:00 a.m. - 78 degrees
3) room temperature - 70 degrees (constant)

You realize that the police believe you to be a prime suspect, so you need an alibi. You know that you were studying with friends until 3:00 a. m., but you aren't sure if that is enough information. You need to know the time of death!

Assuming that the difference between body temperature and room temperature changes at a rate proportional to that difference, and that the normal body temperature is 98.6 degrees, how good is your alibi?

The function that relates an object's temperature to time is a negative exponential function of the form:
body_temp = room_temp + initial_temp * 1/[e^{(cool_rate * time)}]

Yet, we are to assume, in this problem, that the difference between body temperature and room temperature changes at a rate proportional to that difference.

At 10am the body was 78 degrees.
At  9am the body was 80 degrees.

Following this proportional rate of cooling,
At  8am the body was 82.5 degrees.
At  7am the body was 85.6 degrees (approximately).
At  6am the body was 89.5 degrees (approximately).
At  5am the body was 94.4 degrees (approximately).

The time when the body was 98.6 degrees was, then, approximately 4:17am.  Thus, an alibi studying with friends till 3am will not be good enough to erase suspicion.

 

Edited on September 1, 2008, 11:55 am

Posted by Dej Mar on 2008-08-31 22:11:26