Coroners estimate time of death using the rule of thumb that a body cools about 2 degrees F during the first hour after death and about 1 degree F for each additional hour. Assuming an air temperature of 78 degrees F and a living body temperature of 98.6 degrees F, the temperature in degrees F of a body at a time hours since death is given by
1. For what value of will the body cool by 2 degrees F in the first hour?
2. Using the value of found above, after how many hours will the temperature of the body be decreasing at a rate of 1 degree F per hour?
after hours
3. Using the value of found above, show by calculating both values that, 24 hours after death, the coroner's rule of thumb gives approximately the same temperature as the formula.
degrees F
rule of thumb gives T = degrees F
To solve these problems, we need to use the given information and apply the rules provided.
Let's start with the first question:
1. For what value of will the body cool by 2 degrees F in the first hour?
We are given that the body cools about 2 degrees F during the first hour after death. Therefore, the rate of cooling per hour is -2 degrees F.
Let's set up an equation using the formula for the temperature of the body at time hours since death:
T = 78 + (98.6 - 78) - 2h
We want to find the value of h when the body cools by 2 degrees F, so we set T = 2 degrees F:
2 = 78 + (98.6 - 78) - 2h
Simplifying the equation, we can solve for h:
2 = 20.6 - 2h
2h = 20.6 - 2
2h = 18.6
h = 9.3
Therefore, the value of h when the body cools by 2 degrees F in the first hour is approximately 9.3 hours.
Moving on to the second question:
2. Using the value of found above, after how many hours will the temperature of the body be decreasing at a rate of 1 degree F per hour?
We have already found that when h is 9.3, the body cools by 2 degrees F in the first hour. We know that the rate of cooling per hour is -2 degrees F, so after the first hour, the rate of cooling per hour becomes -1 degree F.
Using the same formula as before, we can set up an equation:
T = 78 + (98.6 - 78) - 2h
We want to find the value of hours when the body cools at a rate of 1 degree F per hour, so we set the rate of cooling per hour, -2h, equal to -1:
-2h = -1
h = 0.5
Therefore, after approximately 0.5 hours, the temperature of the body will be decreasing at a rate of 1 degree F per hour.
Finally, let's move on to the third question:
3. Using the value of found above, show by calculating both values that, 24 hours after death, the coroner's rule of thumb gives approximately the same temperature as the formula.
We have found that when h is 9.3, the body cools by 2 degrees F in the first hour. Let's calculate the temperature using both the coroner's rule of thumb and the formula provided.
Coroner's rule of thumb:
Temperature decrease per hour = 1 degree F
Temperature decrease after 24 hours = 1 * 24 = 24 degrees F
Starting from the living body temperature of 98.6 degrees F, the temperature according to the coroner's rule of thumb after 24 hours would be 98.6 - 24 = 74.6 degrees F.
Using the formula from question 1:
T = 78 + (98.6 - 78) - 2h
Substituting h = 24 into the formula:
T = 78 + (98.6 - 78) - 2 * 24
T = 78 + 20.6 - 48
T = 50.6
Therefore, 24 hours after death, the coroner's rule of thumb gives a temperature of approximately 74.6 degrees F, while the formula gives a temperature of approximately 50.6 degrees F.
This shows that the coroner's rule of thumb and the formula do not give the same temperature after 24 hours.