physics Your instructor has been murdered in his office. His body is discovered

Question

physics Your instructor has been murdered in his office. His body is discovered by one of his students at 5pm. The student happens to have a thermometer and finds that the professor?s body temperature s 90 deg. An hour later the police take the temperature of the body and it Is 80 deg. If the office temperature has been maintained at a constant 70 deg. and the professor?s temperature when alive is assumed to be about 100 deg., at what time was your instructor murdered? (Use Newton's Law of Cooling).

Explanation / Answer

temp = T + (I - T)e^(kt)

T = surrounding temperature
I = initial temperature =100 F = 37.7c
k = constant
t = time since cooling started
90 f = 32.2C,100 f = 37.7C, 80 f = 26.6C,70f = 21.1C
The equations are:
at time=t: 32.2 = T + (I - T)ekt
at time=t+1: 26.6= T + (I - T)ekt+1
at time=t+2: 21.1 = T + (I - T)ekt+2

take the first two equations and using I=37.7C we can get an equation with T and k, eliminating the time t.

solve the first for ekt giving
32.2 = T + (37.7 - T)ekt
32 .2- T = (37.7 - T)ekt
ekt = (32.2 - T)/(37.7 - T)

now use this to get ek from the second:
26.6 = T + (37.7 - T)ekt+1
(26.6 - T)/(37.7 - T) = ekt+1ek
(26.6 - T)/(37.7 - T) = ektek
= [(26.6 - T)/(37.7 - T)]ek
(26.6 - T)/(32.2 - T) = ek

and solving the same with equations 1 and 3 to get e2k
:
37.7 = T + (37.7 - T)eK(t+2)
(37.7 - T)/(37.7 - T) = eK(t) e2K

= [(32.2 - T)/(37.7 - T)]e2K
(37.7 - T)/(32.2 - T) = e2K

now take the expression for e^k, square it to get e^(2k) and set it equal to the above left-hand side:
(37.7 - T)/(32.2 - T) = [(26.6 - T)/(32.2 - T)]2
(37.7 - T)(32.2 - T) = (26.6 - T)2
(37.7)(32.2) - 69.9T + T2 = 707.56 - 64.4T + T2

506.38 - 5.5 T = 0

T = 506.38/5.5

T = 92.06
So the surrounding temperature is 21.1C

Next solve for the proportionality constant k.
(26.6 - T)/(32.2 - T) = ek

ek= (26.6 - 92.06)/(32.2 - 92.06) = 1.09

k = 0.86

the time since cooling started.

32.2= T + (37.7 - T)e^(kt)
32.2 = 92.06 + (37.7 - 92.06)e0.86t
59.86/54.36 = e0.86t
t = ln(1.10)/(0.86)
t = 0.23h ,

t = 13.8 mins

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