Need an Alibi

[BMAD]

Suppose that 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 quickly realize that the police believe you to be a prime suspect, so you need an alibi. You know that you were studying until midnight, prove your innocence or go to jail.

Edited September 3, 2015 by BMAD

[jasen]

2 Alibi

1. Chalk dust is the key. When I meet the Professor, I'm clean, no chalk dust anymore on my body, so I'm not the killer.

2. If I'm the killer, I will not call the police. I will go immediately, so nobody knows the killer.

[BMAD]

2 Alibi

1. Chalk dust is the key. When I meet the Professor, I'm clean, no chalk dust anymore on my body, so I'm not the killer.

2. If I'm the killer, I will not call the police. I will go immediately, so nobody knows the killer.

2 very simple choices a killer could do if they wanted to confuse the detectives. 

[Rainman]

What part of the body was measured for temperature? Different parts of the body cool at different rates.

[BMAD]

What part of the body was measured for temperature? Different parts of the body cool at different rates.

assume in this case that it doesn't matter

[harey]

 

The body temperature decreases exponentially. This means 84 at 8:00, 92 at 7:00, 100 somewhere before 6:30.

As you do not study from books anymore but online, various logs will show active connection from my room.

 

Edited September 6, 2015 by harey
forgot spoiler

[BMAD]

 

Hidden Content

On the right track but I have a different time.

[harey]

On the right track but I have a different time.

I calculated very roughly. Interpolating an exponential is quite dangerous. If the temperature was rounded by 0.2 degrees and measured 2-3 minutes before/after the whole hour, you can get quite a different result.

Proof is left to the reader.

[BMAD]

Consider this a differential equation problem.

[plasmid]

Define the terms
T = temperature (degrees above room temp)
t = time (hours from 10 AM)
c = a constant describing the rate of cooling

If the rate of cooling is proportional to the temperature gradient, then
dT/dt = -cT

The only way (I know of) to have a function's derivative be itself is to use some form of e^x. In this case, we can have
T = e^-ct
let u = -ct while solving the derivative so
T = e^u
dT/dt = dT/du * du/dt = e^u * -c
dT/dt = -c * e^-ct = -cT

So a solution is
T = e^-ct + C

At 10 AM (time t = 0) the body temp is 78 degrees (T = 8 degrees above room temp), so
T = e^-ct + C
8 = e⁰ + C
C = 7

At 9 AM (time t = -1) the body temp is 80 degrees (T = 10), so
T = e^-ct + C
10 = e^c + 7
c = ln(3)

Then solve for when T = 28.6 (98.6 - 70 degrees)
T = e^-ct + C
28.6 = e^{-t ln(3)} + 7
ln(21.6) = -t ln(3)
-t = 2.797 hours (from 10 AM) = 7:12

Where were you at 7:12 AM? I have no idea.

[BMAD]

the statement tells you that the rate of change (also known as the time derivative) of T is proportional to T itself. Thus, you would write dT/dt = -k*T, where dT/dt stands for the time derivative and k is the constant of proportionality. The reason for the minus sign is that T is always positive but decreasing, and thus, T > 0, but dT/dt < 0.

Edited September 11, 2015 by BMAD

[BMAD]

Hidden Content

lol, forgot the statement: 

The difference between body temperature and the room temperature changes at a rate proportional to that difference.