It was noon on a cold December day in Tampa: 16°C. Detective Taylor arrived at the crime scene to find the sergeant leaning over the body. The sergeant said there were several suspects. If they knew the exact time of death, then they could narrow the list. Detective Taylor took out a thermometer and measured the temperature of the body: 34.5°C. He then left for lunch. Upon returning at 1:00 p.m., he found the body temperature to be 33.7°C. When did the murder occur? [Hint: Normal body temperature is 37°C.]
The murder was occurred at 9:08 a.m.
It was noon on a cold December day in Tampa: 16°C. The initial temperature of the body measured by detective Taylor in the noon is 34.5°C. At 1:00 p.m., the temperature of the body was found to be 33.7°C, which means the temperature of the body after 1 hour was 33.7°C. By using Newton’s law of cooling, we have to determine the time at which the murder occurred, i.e., the time at which the temperature of the body was 37°C.
Newton’s Law of Cooling is,
Here, we will take the values as,
Initial temperature of the body,,
Temperature in Tampa in the noon,
Temperature of the body after 60 min,
Using the given values in equation (1), to find the value of k,
We will use this value of k in next step to find the time at which the temperature of the body was 37°C.
Substituting in equation (1),
Here, negative sign indicates that the temperature of the body was 37°C, 171 minutes before the noon.
Hence, the murder was occurred at 9:08 a.m.
A garage with no heating or cooling has a time constant of 2 hr. If the outside temperature varies as a sine wave with a minimum of at and a maximum of at , determine the times at which the building reaches its lowest temperature and its highest temperature, assuming the exponential term has died off.
On a mild Saturday morning while people are working inside, the furnace keeps the temperature inside the building at 21°C. At noon the furnace is turned off, and the people go home. The temperature outside is a constant 12°C for the rest of the afternoon. If the time constant for the building is 3 hr, when will the temperature inside the building reach 16°C? If some windows are left open and the time constant drops to 2 hr, when will the temperature inside reach 16°C?
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