Suppose that you are a CSI and you are called to a crime scene at 12:00 noon and find a dead body in a walk-in that is a constant 40¨H. The initial temperature is 85¡ã and after 20 minutes. The body¡¯s temperature is 84¡ã. Using Newton¡¯s Law of cooling u(t) = T +(u0 ¨C T)ek . t , k< 0, T temperature of the surrounding medium, u0 is the initial temperatures of the object. If we assume that 98.6 ¨H when was the time of death?
To determine the time of death using Newton's Law of Cooling, we need to use the formula:
u(t) = T + (u0 - T)e^(k*t)
where:
- u(t) is the temperature of the body at time t
- T is the temperature of the surrounding medium (40°F)
- u0 is the initial temperature of the body (85°F)
- k is the cooling constant (which we need to find)
- t is the time elapsed since death (which we want to find)
Given that the body's temperature after 20 minutes is 84°F, we can calculate k as follows:
84 = 40 + (85 - 40)e^(k*20)
Subtracting 40 from both sides:
44 = 45e^(k*20)
Dividing both sides by 45:
e^(k*20) = 44/45
Now, take the natural logarithm (ln) of both sides to isolate k:
k*20 = ln(44/45)
Dividing both sides by 20:
k = ln(44/45)/20
Calculating k:
k ≈ -0.0031
Now that we have the cooling constant, we can find the time of death when the body's temperature reached 98.6°F by rearranging the formula:
98.6 = 40 + (85 - 40)e^(-0.0031 * t)
Subtracting 40 from both sides:
58.6 = 45e^(-0.0031 * t)
Dividing both sides by 45:
e^(-0.0031 * t) = 1.3022
Taking the natural logarithm (ln) of both sides:
-0.0031 * t ≈ ln(1.3022)
Dividing both sides by -0.0031:
t ≈ ln(1.3022) / -0.0031
Calculating t:
t ≈ -224.7 minutes
Since negative time is not possible in this case, it means that the body temperature did not reach 98.6°F, indicating that the person died before reaching this temperature. Therefore, we cannot determine the exact time of death based on the provided information and assumptions.