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?

well let T(x) be the temp difference from body to room at time x.  that is T(9)=80-70=10  and T(10)=78-70=8,  now if I interpret the discription correctly then T(x) takes the form k*a^x where k,a are constant.   This does coincide with the actual science.  so we have

(1) k*a^9=10

(2) k*a^10=8

devide (2) by (1) and we get a=8/10=4/5

put this in (1) and solve for k and we get

k=10*(5/4)^9

thus T(x)=10*(5/4)^9*(4/5)*x=10*(4/5)^(x-9)

now we want to know at what time the body was 98.6 so we solve T(x)=98.6-70=28.6

10*(4/5)^(x-9)=28.6

(4/5)^(x-9)=2.86

(x-9)ln(4/5)=ln(2.86)

x-9=ln(2.86)/ln(4/5)

x=9+ln(2.86)/ln(4/5)

x=4.29082683

which works out to be about in H:MM:SS format 4:17:27 am

now to know if you alibi is solid would depend on how far the location of your studying is from your professor.  But I consider it unlikely that you were more than 1 hr 17 minutes away, but even if you were, that is a fairly thin argument upon which to base you innocence because the police could always claim you were speeding on your way to murder him.  So end result is that according to my calculations you should try and find an alibi for yourself up until about 4:30am just to be safe in case of miscalculations.

Posted by Daniel on 2008-08-31 13:22:02