The body of a murder victim was discovered at 11:00 A.M. and found the temperature of the body was 94.60°F. The temperature of the room was 700°F. One hour later, in the same room, he took the body temperature again and found it was 93.40°F. Estimate the time of death.
Are you taking Calculus?
Hint: Newton's Law of Cooling
look at the second question below in Related Questions answered by Mathmate in 2011
http://www.jiskha.com/display.cgi?id=1304625732
Well, this sounds like quite the heated situation! Let's cool down the facts and solve this mystery, shall we?
If the room temperature was a scorching 700°F, I gotta say, it sounds like there was more than just a murder going on. It seems like someone turned the room into a sauna!
Now, if we look at the body temperature, we see that it dropped from 94.60°F to 93.40°F in one hour. That's a decrease of 1.20°F.
Given that the average rate at which the body loses heat is about 1.5°F per hour, we can make a good guess that the time of death was around 2 hours prior to the discovery of the body at 11:00 A.M.
But hey, remember that I'm just a funny little bot, not a forensic expert. If it's a serious situation, I strongly recommend consulting the professionals who can give you a more precise and accurate answer. Stay safe out there!
To estimate the time of death, we can calculate the cooling rate of the body using Newton's law of cooling, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.
First, let's calculate the cooling constant (k) using the temperature measurements taken one hour apart.
Change in temperature = Initial temperature - Final temperature
Change in temperature = 94.60°F - 93.40°F = 1.20°F
Change in time = 1 hour
k = (natural log of change in temperature) / (change in time)
k = ln(1.20) / 1 ≈ 0.1823
Now, let's use the cooling constant to estimate the time of death. Since the body was found at 11:00 A.M., we'll work with a 24-hour clock.
Step 1: Calculate the temperature difference between the body and the room temperature at the time of discovery.
Temperature difference = Initial body temperature - Room temperature
Temperature difference = 94.60°F - 70°F = 24.60°F
Step 2: Calculate the temperature difference between the final body temperature and the room temperature.
Temperature difference = Final body temperature - Room temperature
Temperature difference = 93.40°F - 70°F = 23.40°F
Step 3: Calculate the time it took for the temperature to decrease by the temperature difference.
Time difference = (natural log of temperature difference at discovery / temperature difference at final measurement) / cooling constant
Time difference = (ln(24.60) - ln(23.40)) / 0.1823 ≈ 0.0665 hours
Step 4: Estimate the time of death by subtracting the time difference from the time the body was discovered.
Time of death = 11:00 A.M. - 0.0665 hours = approximately 10:59 A.M.
Therefore, based on the given information and calculations, the estimated time of death is around 10:59 A.M.
To estimate the time of death, we can use the concept of body cooling known as the Newton's law of cooling. According to this law, the rate of body cooling can be approximated by an exponential decay function.
The equation for Newton's law of cooling is:
T(t) = T(r) + (T(0) - T(r)) * e^(-kt)
Where:
- T(t) is the temperature of the body at time 't'
- T(r) is the temperature of the room
- T(0) is the initial temperature of the body
- k is the cooling constant
- e is the base of the natural logarithm
We know the following information from the question:
- At 11:00 A.M., the temperature of the body was 94.60°F.
- The room temperature was 70°F.
- One hour later, at 12:00 P.M., the temperature of the body was 93.40°F.
Using this information, we can set up the following equations:
94.60 = 70 + (T(0) - 70) * e^(-k*1)
93.40 = 70 + (T(0) - 70) * e^(-k*2)
Simplifying these equations, we get:
e^(-k) = (94.60 - 70) / (T(0) - 70)
e^(-2k) = (93.40 - 70) / (T(0) - 70)
Dividing these two equations, we can eliminate the variable 'T(0)' and solve for 'k':
(e^(-k)) / (e^(-2k)) = [(94.60 - 70) / (T(0) - 70)] / [(93.40 - 70) / (T(0) - 70)]
Simplifying further, we have:
e^k = (94.60 - 70) / (93.40 - 70)
Taking the natural logarithm of both sides, we get:
k = ln((94.60 - 70) / (93.40 - 70))
Calculating this value using a calculator or software, we find:
k ≈ 0.0499 (rounded to four decimal places)
Now we can substitute the value of 'k' back into one of the original equations to solve for 'T(0)':
94.60 = 70 + (T(0) - 70) * e^(-0.0499*1)
Rearranging the equation:
(T(0) - 70) * e^(-0.0499*1) = 94.60 - 70
Simplifying:
(T(0) - 70) * e^(-0.0499) = 24.60
Dividing by e^(-0.0499):
T(0) - 70 = 24.60 / e^(-0.0499)
Calculating this value, we find:
T(0) ≈ 94.9465°F (rounded to four decimal places)
Therefore, the estimated initial body temperature (T(0)) was approximately 94.9465°F.
To estimate the time of death, we need to determine how long it took for the body to cool from the initial temperature to the temperature measured at 11:00 A.M. (94.60°F).
Using the equation T(t) = T(r) + (T(0) - T(r)) * e^(-kt), we can rearrange it to solve for 't':
94.60 = 70 + (94.9465 - 70) * e^(-0.0499*t)
Simplifying:
24.60 = 24.9465 * e^(-0.0499*t)
Dividing by 24.9465:
e^(-0.0499*t) = 1
Taking the natural logarithm of both sides, we get:
-0.0499*t = ln(1)
Since ln(1) = 0, we can solve for 't':
-0.0499*t = 0
t = 0
Therefore, it took 0 hours for the body to cool from the initial temperature to the temperature measured at 11:00 A.M.
Since the body temperature was measured at 11:00 A.M., we can conclude that the estimated time of death is approximately one hour prior, at 10:00 A.M.