A body was found at $6 a.m.$ in a warehouse where the temperature was $50^{\circ} F$. The medical examiner found the temperature of the body to be $66^{\circ} F$. What was the approximate time of death? Use Newton's law of cooling, with $k=$ 0.1947 .
$T(t)=T_A+\left(T_o-T_A\right) e^{-k t}$
A. Midnight (12 a.m.)
B. 5 a.m.
C. 2 a.m.
D. 3 a.m.
Answer (2)
Apply Newton's Law of Cooling: T ( t ) = T A + ( T 0 − T A ) e − k t .
Substitute given values: T A = 50 , T ( t ) = 66 , T 0 = 98.6 , and k = 0.1947 .
Solve for t : 66 = 50 + ( 98.6 − 50 ) e − 0.1947 t , which simplifies to t ≈ 5.73 hours.
Subtract t from the discovery time (6 a.m.) to estimate the time of death, resulting in approximately Midnight (12 a.m.) .
Explanation
Understanding Newton's Law of Cooling We are given the formula for Newton's Law of Cooling: T ( t ) = T A + ( T 0 − T A ) e − k t , where:
T ( t ) is the temperature of the body at time t ,
T A is the ambient temperature, T 0 is the initial temperature of the body (at the time of death), k is the cooling constant, and t is the time elapsed since death.
Identifying Given Values We are given:
T A = 5 0 ∘ F (ambient temperature), T ( t ) = 6 6 ∘ F (temperature of the body at 6 a.m.), T 0 = 98. 6 ∘ F (normal body temperature), k = 0.1947 (cooling constant).
We want to find the time t elapsed since death.
Plugging in the Values Plug the given values into the formula:
66 = 50 + ( 98.6 − 50 ) e − 0.1947 t
Simplify the equation:
66 = 50 + 48.6 e − 0.1947 t 16 = 48.6 e − 0.1947 t
Isolating the Exponential Term Solve for t :
48.6 16 = e − 0.1947 t 486 160 = e − 0.1947 t 243 80 = e − 0.1947 t
Applying Natural Logarithm Take the natural logarithm of both sides:
ln ( 243 80 ) = − 0.1947 t
Calculating Time Elapsed Solve for t :
t = − 0.1947 l n ( 243 80 )
Using a calculator, we find that:
t ≈ − 0.1947 − 1.1163 ≈ 5.73 hours
Determining the Time of Death The body was found at 6 a.m. We need to subtract 5.73 hours from 6 a.m. to find the time of death.
6 a.m. − 5.73 hours = 6 : 00 − 5 : 44 = 0 : 16 ≈ 12 : 16 a.m.
So, the approximate time of death is around 12:16 a.m.
Final Answer Choice Therefore, the closest answer is Midnight (12 a.m.).
Examples
Newton's Law of Cooling is not just a theoretical concept; it has practical applications in forensics, cooking, and engineering. For example, forensic scientists use it to estimate the time of death by analyzing the temperature of a body. Chefs use it to predict how long it will take for a hot dish to cool down to a safe serving temperature. Engineers apply it to design cooling systems for electronic devices, ensuring they don't overheat. Understanding this law helps in various real-world scenarios where temperature change over time is a crucial factor.
Using Newton's Law of Cooling, we calculated the time elapsed since death to be approximately 5.68 hours before the body was discovered at 6 a.m., estimating the time of death to be around midnight (12 a.m.). Therefore, the answer is A. Midnight (12 a.m.).
;
