On a cold December day in Dallas, Detective Daniels went to an apartment complex to investigate a murder.
When he arrived at noon, the sergeant informed the detective that they were having trouble determining the
time of death. Detective Daniels measured the temperature of the body, finding it to be 77.9℉. He also noted
that the thermostat in the room was set at 72℉. He then left for lunch, announcing that when he returned, he
would tell them when the murder was committed. Upon his return at 1:00PM, he found the body temperature
to be 75.6℉.
At first it looks like Detective Daniels doesn’t have enough information to find the time of death. However,
Detective Daniels knows Newton’s Law of Cooling, which can be used to predict the time for an object to cool to
a given temperature.
() = + (0 − )−
() is the temperature of the object at time , is the temperature of the surrounding environment, 0 is the
initial temperature of the object, and is the cooling rate.
a. Using the temperatures of the body observed over one hour, along with the temperature of the room, find
the cooling rate, . Round to five decimal places.
b. Write the cooling function, (), using the fact that the initial temperature of the body, 0, was 98.6℉.
c. Let () = 77.9℉. Use the cooling function from part b, (), to solve for , the number of hours since the
body was murdered. Around what time was the murder committed?