# Newton's Law of Cooling

Case: At 9:00 PM a coroner arrived at a hotel room of a murder victim. The temp. of the room was 70 degrees F. It was assumed that the victim had a body temp. of 98.6 degrees F (AT THE TIME OF DEATH)(not at 9:00PM). The coroner took the victim's temp. at 9:15 PM at which it was 83.6 degrees F and again at 10:00 PM at which it was 80.3 degrees F.

A. At what time did the victim die? I solved that question to be at 7:15 PM (165 min before 10:00PM)

B. If the assumption of the victim's body temp. at the time of death was found to be incorrect due to a major outbreak of influenza at the hotel during the victim's stay and the usual fever a flu case will run is 103 degrees F. What is the time of death then? I solved it to be 6:37 PM(203 min before 10:00PM).

C. If the victim had no influenza at the time of death, but the room temp was really 75 degrees F what was the time of death then? answer was 7:47PM(133 min before 10:00PM)

D. Assuming the victim was murdered 3 minutes of the time the murderer got off the elevator(which had video surveillance) what five minute sequence should have been reviewed in each scenarios a, b, and ,c?

keywords: Newtons

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#### Solution Preview

Case A:

In this case, the logarithmic rate of cooling per minute is

rateA = log( (83.6-70)/(80.3-70) ) / 45 = 0.0062 (1/min),

where "log" denotes the natural logarithm,

The time needed to cool from 96.6F to 80.3F is

TimeCoolA = log( (96.6-70)/(80.3-70) ) / rateA = 154 min

The murder time is

timeMurderA = 10:00 pm - 154 min = ...

#### Solution Summary

Newton's Law of Cooling is investigated.