A dead body was found within a closed room of a house where the temperature was a constant 21° C. At the time of discovery the core temperature of the body was determined to be 27° C. One hour later a second measurement showed that the core temperature of the body was 25° C. Assume that the time of death corresponds to t = 0 and that the core temperature at that time was 37° C. Determine how many hours elapsed before the body was found. [Hint: Let t1 > 0 denote the time that the body was discovered.] (Round your answer to one decimal place.)
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Expert's answer
2020-06-23T14:23:03-0400
T′=k(T−Tm)(Newton’s Law of Cooling)∴∫T−TmdT=∫kdt∴ln∣T−Tm∣=kt+C1→T(t)=Tm+cekt∵T(closed room)=21→Tm=21∵T(temperature at t=0 )=37→T(0)=37Substitute in T(t)=Tm+cekt∴37=21+ce0→c=16∴T=21+16ekt(1)∵Where t is the time at discovery ∵The temperature was 27∘at the time of discovery∴27=21+16ekt→16ekt=6(2)∵The temperature was 25∘after one hour(t+1)∴25=21+16ek(t+1)→16ek(t+1)=4(3)∵Dividing (2) by (3)ek(t+1)ekt=46=23ekt−kt−k=23→e−k=23→−k=ln23→k=−0.41Substitute in (2) 16e−0.41t=6→e−0.41t=166=83∴−0.41t=ln83∴t=ln83÷(−0.41)∴t=2.4hours
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