Question

Suppose the temperature of a body when discovered is

85 F. Two hours later, the

temperature is

74 F and the room temperature is

68 F. Find the time when the body was

discovered after death (assume the body temperature to be

98 6. F at the time of death.)

Answer 1

At the death time temperature is 98 6 F

Answer 2

A body was found at midnight and it was 80 degrees. 2 hours later, it was 75 degrees. The room is 60 degrees. What time did the body die



Let   t = time in hours since the body died.  And t = tm is hours after death at midnight.

   T(t) = The body temperature as a function of time after death (in degrees F)

      To = Body temperature at instant of death or at time zero (= 98.6oF)

      Ts = Constant temperature of the surroundings (the morgue) = 60oF

       K = A constant in the exponential decay of the body's temperature (to be determined)

 

Then Newton's formula is :  T(t) = Ts+ (To - Ts) e (-Kt)  =  60 + (98.6 - 60) e (-Kt)    = 60 + 38.6 e (-Kt) .

 

Apply the known data:

 1.   T(tm) = 80 = 60 + 38.6 e (-Ktm) which simplified becomes  e (-Ktm)  = 20/38.6 = 0.51813,

             so -Ktm = lne (0.51813) = - 0.65752  or     Ktm = 0.65752.

 2.   T(tm + 2) = 75 = 60 + 38.6 e (-K[tm+ 2]),  which simplified becomes  e (-K[tm + 2]) = 15/38.6 = 0.38860,

             so (-K[tm + 2]) = lne (0.38860) = - 0.94520  or K[tm + 2] = 0.94520  

 

Two equations with two unknowns, so solve for K and tm.

Rewrite the second one as  2K = 0.94520 - Ktm = 0.94520 - 0.65752 = 0.28768 , K = 0.14384

And from the first equation  tm = 0.65752/K = 0.65752/0.14384 = 4.57 hours

 

By Newton's model of temperature loss, the body died about 4.57 hours BEFORE midnight since by midnight its temperature had already dropped from a living person at 98.6oF to 80oF.



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