(An Alibi problem) Newton's law of cooling system state's that if an object at temperature

Question

1. (An Alibi problem) Newton's law of cooling system state's that if an object at temperature T0 is placed into an environment that has constant temperature Te, then the temperature of the object, T(t), after T minutes is given by T(t)=Te+(T0-Te)e^-kt, where k is a positive constant. Josh has a college Algebra exam at a state university from 7.00pm-8.30pm on Oct. 27, 2005. The outside temperature of this particular evening was 62.6 degree F. at 9.00pm the campus police found Josh staring at a body lying on the grass. The police immediately took the temperature of the body of the victim it was 86.6 degree F. The police suspected that josh killed the victim. At 10.pm the state police arrived and  measure the temperature of the body of the victim to be 78.6 degree F. Assuming that the average normal body temperature is 98.6 degree F, and that Josh came out of the exam room at 8.31pm,use the Newton's law of cooling to determine whether Josh was innocent or not.  

Answer 1

T(t) = 62.6 + (86.6 - 62.6)e^-kt

 

       = 62.6 + 24e^-kt

 

       = temperature of body t minutes after 8:31 PM

 

T(89) = temperature of body at 10 PM

 

         = 62.6 + 24e^-89k = 78.6

 

                        24e^-89k = 16

 

                           e^-89k = 0.6666666

 

                            -89k = ln(0.6666666)

 

                                k = -0.0045558

 

T(t) = 62.6 + 24e^-0.0045558t

 

To find the time of death, solve the equation T(t) = 98.6:

 

   62.6 + 24e^-0.0045558t = 98.6

 

              24e^-0.0045558t = 36

 

                  e^-0.0045558t = 1.5

 

                   -0.0045558t = ln1.5

 

                                   t = -88.99976 minutes

 

The time of death was 88.99976 minutes before 8:31 PM, or at about 7:02 PM.  Assuming that Josh was on time for his exam, he would have been taking the exam at 7:02 PM, so he could not have committed the crime.