I need help...especially finding out the time...
At 1:00 a.m., a homicide detective found the reading of a corpse's body temperature to be 88 degrees Fahrenheit. One hour later, the body temperature is 83 degrees Fahrenheit. If the body has been in a 74 degree Fahrenheit room since its death, what is the time of death assuming the body was at 98.6 degrees Fahrenheit at its death?
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To determine the time of death, we can use the concept of body cooling to estimate the elapsed time based on the change in body temperature.
First, let's calculate the cooling rate of the body. The formula to calculate body cooling is Newton's law of cooling:
T = T_r + (T_i - T_r) × e^(-kt)
Where:
T = Final body temperature
T_i = Initial body temperature
T_r = Room temperature
k = Cooling constant
t = Time in hours
In this case, we are given the following information:
- Initial body temperature (T_i) = 98.6 degrees Fahrenheit
- Room temperature (T_r) = 74 degrees Fahrenheit
To find the cooling constant (k), we can use the formula:
k = ln((T_i - T_r) / (T - T_r)) / t
Let's calculate the cooling constant:
k = ln((98.6 - 74) / (88 - 74)) / 1
k = ln(24 / 14) / 1
k = ln(1.71) ≈ 0.537
Now, let's use the cooling equation to find the time of death.
We are given the following information:
- Initial body temperature (T_i) = 98.6 degrees Fahrenheit
- Body temperature after one hour (T) = 83 degrees Fahrenheit
- Room temperature (T_r) = 74 degrees Fahrenheit
- Cooling constant (k) ≈ 0.537
Plugging these values into the equation:
83 = 74 + (98.6 - 74) × e^(-0.537t)
Simplifying the equation:
9 = 24e^(-0.537t)
Divide both sides by 24:
9/24 = e^(-0.537t)
0.375 = e^(-0.537t)
To solve for t (time in hours), we need to use the natural logarithm:
ln(0.375) = -0.537t
Using a calculator:
t ≈ ln(0.375) / -0.537 ≈ 2.77 hours
Therefore, the estimated time of death would be approximately 1:00 a.m. minus 2.77 hours, which is around 10:15 p.m.