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?
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?
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Chapter1: Units, Trigonometry. And Vectors
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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?
Newton's 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:
\[ T(t) = T_e + (T_0 - T_e) e^{-kt} \]
where \( T(t) \) is the temperature of the object at time \( t \), \( T_e \) is the constant temperature of the environment, \( T_0 \) 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:
\[ \frac{T(t) - T_e}{T_0 - T_e} = e^{-kt} \]
**Tip:** To organize our thinking about this problem, let’s 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c32cba2-8623-416e-958a-026251d13dab%2F51aaf0bc-2216-4ddf-afab-be7ccdbcd02b%2Fn6o6sr_processed.jpeg&w=3840&q=75)
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?
Newton's 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:
\[ T(t) = T_e + (T_0 - T_e) e^{-kt} \]
where \( T(t) \) is the temperature of the object at time \( t \), \( T_e \) is the constant temperature of the environment, \( T_0 \) 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:
\[ \frac{T(t) - T_e}{T_0 - T_e} = e^{-kt} \]
**Tip:** To organize our thinking about this problem, let’s 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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