A detective discovers a murder victim in a room at the Marriott Marquis Hotel at 9:15 pm on Friday night. Immediately, the temperature of the body is recorded as being 78 °F. The programmable thermostat has been set to 70 °F for the last week. What was the time of death?

Principles of Physics: A Calculus-Based Text
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Chapter17: Energy In Thermal Processes: The First Law Of Thermodynamics
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A detective discovers a murder victim in a room at the
Marriott Marquis Hotel at 9:15 pm on Friday night.
Immediately, the temperature of the body is recorded
as being 78 °F. The programmable thermostat has
been set to 70 °F for the last week. What was the time
of death?
Newtons Law of Cooling is an exponential equation,
which describes the cooling of a warmer object to the
cooler temperature of the environment. Specifically,
we write this law as,
T(t) = Te+(To - T)e¬kt
%3D
|
where T (t) is the temperature of the object at
time t, Te is the constant temperature of the
environment, To is the initial temperature of the object,
and k is a constant that depends on the material
properties of the object. To solve this exponential
equation for t, you will need to use logarithms. This
equation can be rearranged to:
T (t) - Te
e-kt
To -Te
Tip: To organize our thinking about this problem, lets
be explicit about what we are trying to solve for. We
would like to know the time at which a person died. In
particular, the investigator arrived on the scene at 9:15
pm, which is t hours after death. The temperature of
the body was found to be 78 °F. Assume k = 0.1335
and the victim's body temperature was normal (98.6
°F) prior to death. Show all work
Transcribed Image Text:A detective discovers a murder victim in a room at the Marriott Marquis Hotel at 9:15 pm on Friday night. Immediately, the temperature of the body is recorded as being 78 °F. The programmable thermostat has been set to 70 °F for the last week. What was the time of death? Newtons Law of Cooling is an exponential equation, which describes the cooling of a warmer object to the cooler temperature of the environment. Specifically, we write this law as, T(t) = Te+(To - T)e¬kt %3D | where T (t) is the temperature of the object at time t, Te is the constant temperature of the environment, To is the initial temperature of the object, and k is a constant that depends on the material properties of the object. To solve this exponential equation for t, you will need to use logarithms. This equation can be rearranged to: T (t) - Te e-kt To -Te Tip: To organize our thinking about this problem, lets be explicit about what we are trying to solve for. We would like to know the time at which a person died. In particular, the investigator arrived on the scene at 9:15 pm, which is t hours after death. The temperature of the body was found to be 78 °F. Assume k = 0.1335 and the victim's body temperature was normal (98.6 °F) prior to death. Show all work
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