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?

To determine the time of death, we can use the concept of the rate at which a body cools after death, known as the Newton's law of cooling.

According to Newton's law of cooling, the rate of change of the body's temperature is directly proportional to the difference between the body's temperature and the ambient (room) temperature.

Let's assume:
- Td = temperature of the body at death (Td = 98.6 degrees Fahrenheit)
- Tr = ambient (room) temperature (Tr = 74 degrees Fahrenheit)
- T1 = temperature of the body at 1:00 a.m. (T1 = 88 degrees Fahrenheit)
- T2 = temperature of the body at 2:00 a.m. (T2 = 83 degrees Fahrenheit)
- k = cooling constant

Using these values, we can write two equations based on Newton's law of cooling.

1. At 1:00 a.m.:
T1 - Tr = (Td - Tr) * e^(-k * t1)

2. At 2:00 a.m.:
T2 - Tr = (Td - Tr) * e^(-k * t2)

Subtracting equation 1 from equation 2 to eliminate the cooling constant (k) gives:

(T2 - Tr) - (T1 - Tr) = (Td - Tr) * (e^(-k * t2) - e^(-k * t1))

Now, let's substitute the given values:

(83 - 74) - (88 - 74) = (98.6 - 74) * (e^(-k * 2) - e^(-k * 1))

9 - 14 = 24.6 * (e^(-2k) - e^(-k))

-5 = 24.6 * (e^(-2k) - e^(-k))

Dividing both sides by 24.6:

-5 / 24.6 = e^(-2k) - e^(-k)

We can simplify further by letting x = e^(-k):

-0.203 = x^2 - x

Rearranging the equation:

x^2 - x - 0.203 = 0

Solving this quadratic equation, we find:

x ≈ 1.255 or x ≈ -0.543

Since the value of x cannot be negative, we take x ≈ 1.255, which means e^(-k) ≈ 1.255.

Now, we can solve for the time of death (t):

t = ln(1.255) / -k

To find the value of k, we can use the following relation:

e^(-k) ≈ 1.255

Taking the natural logarithm (ln) of both sides:

-ln(k) ≈ ln(1.255)

Solving for k:

k ≈ -ln(1.255)

Substituting the value of k in the equation for t:

t ≈ ln(1.255) / ln(1.255)

Evaluating this expression, we find:

t ≈ 0.693 / -ln(1.255)

Therefore, the approximate time of death is t hours before 1:00 a.m.

To determine the time of death, we can use a formula called the Glaister Equation. The equation allows us to estimate the time of death by comparing the body's temperature with the ambient (room) temperature.

Glaister Equation: T(d) = Ta + ( To - Ta ) × (0.78)^(t / k)

Where:
T(d) = Body temperature at the time of death
Ta = Ambient (room) temperature
To = Initial body temperature
t = Time that has elapsed
k = Cooling constant (which is approximately 1.5 for our purposes)

Given information:
To = 98.6°F (initial body temperature)
Ta = 74°F (ambient temperature)
t = 2 hours (as one hour has passed from the initial time of 1:00 a.m.)

By substituting the values into the equation, we can calculate the estimated temperature at the time of death (T(d)):

T(d) = 74 + (98.6 - 74) × (0.78)^(2 / 1.5)

Now we can compute the estimated temperature at the time of death.

T(d) = 74 + (24.6) × (0.78)^(1.33)
T(d) = 74 + (24.6) × (0.479)
T(d) ≈ 74 + 11.776
T(d) ≈ 85.776°F

According to the calculations, the estimated body temperature at the time of death was approximately 85.776°F.

Since the recorded temperature at 1:00 a.m. was 88°F, which is higher than the estimated temperature at the time of death, we can conclude that the time of death occurred before 1:00 a.m.