Question

A dead body is found at \(10 \mathrm{PM}\). to have temperature \(82^{\circ} \mathrm{F}\). One hour later the temperature was \(76^{\circ} \mathrm{F}\). The temperature of the room was a constant \(70^{\circ} \mathrm{F}\). Assuming that the temperature of the body was \(98.6^{\circ} \mathrm{F}\) when it was alive, estimate the time of death.

Short answer

The estimated time of death was approximately 8:42 PM.

Step-by-step solution

Newton's Law of Cooling Equation

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature. The formula for this is:\[ T(t) = T_a + (T_0 - T_a) e^{-kt} \]where \( T(t) \) is the temperature at time \( t \), \( T_a \) is the ambient temperature, \( T_0 \) is the initial temperature, and \( k \) is a constant.

Setting Up the Problem

We know: - \( T_a = 70^{\circ}\mathrm{F} \)- Initial temperature when the body was alive \( T_0 = 98.6^{\circ}\mathrm{F} \)- Temperature after 0 hours \( T(0) = 82^{\circ}\mathrm{F} \)- Temperature after 1 hour \( T(1) = 76^{\circ}\mathrm{F} \).We need to determine \( t \) when the body temperature was last \(98.6^{\circ}\mathrm{F} \).

Calculate the Constant \( k \)

To find \( k \), use the temperature after 1 hour equation: \[ 76 = 70 + (82 - 70) \cdot e^{-k \cdot 1} \]Simplify to: \[ 6 = 12 \cdot e^{-k} \]Divide by 12: \[ 0.5 = e^{-k} \]Take the natural log on both sides: \[ -k = \ln(0.5) \]\[ k = -\ln(0.5) \approx 0.693 \].

Estimate Time of Death

Using the value of \( k \), find \( t \) when the body temperature was alive (\( T_0 = 98.6^{\circ}\mathrm{F} \)):\[ 82 = 70 + (98.6 - 70) \cdot e^{-kt} \]\[ 12 = 28.6 \cdot e^{-0.693t} \]\[ e^{-0.693t} = \frac{12}{28.6} \approx 0.419 \]Take the natural logarithm:\[ -0.693t = \ln(0.419) \]\[ t \approx \frac{\ln(0.419)}{-0.693} \approx 1.29 \] hours. The body was alive approximately 1.29 hours before being found at 10 PM.

Key concepts

Temperature Estimation

Temperature estimation is an important aspect of solving problems related to Newton's Law of Cooling. In the given problem, we need to estimate the time of death based on the cooling process of a dead body. Temperature estimation involves using known temperatures at specific times to predict unknown values, such as the time the body was last at the normal temperature of a living person, which is 98.6°F.
Newton's Law of Cooling provides a mathematical formula that relates the rate of temperature change of a body to the difference between its current temperature and the ambient temperature. This relationship helps us predict past, present, or future temperatures by evaluating how the temperature changes over time.
The given exercise allows us to estimate the time of death by observing the temperature transitions from the time the body was found till the time it was alive. This kind of estimation is crucial in forensic science, as it helps authorities determine the time of an incident by observing temperature changes over time, using historical temperature data.

Ambient Temperature

In the context of Newton's Law of Cooling, ambient temperature refers to the temperature of the surrounding environment where the body or object of interest is located. This temperature acts as a baseline for evaluating the cooling process.
In our problem, the room where the body was found has an ambient temperature of 70°F, which remains constant throughout the exercise. This constant value is crucial as it signifies the stable environment against which the temperature of the cooling body is compared.
Ambient temperature affects the rate at which a body cools, with higher differences between the body's temperature and the ambient temperature leading to a faster rate of cooling. Understanding the role of ambient temperature allows us to plug it into Newton's Law of Cooling equation effectively to predict temperature changes and make accurate estimations of unknown variables like the time of death in this scenario.

Natural Logarithm

The natural logarithm is a special mathematical function that is particularly useful when dealing with exponential decay or growth, such as those described in Newton's Law of Cooling. In our calculation, the natural logarithm helps us solve for the constant rate "k" and the time of cooling.
To find "k", we use the equation for the temperature at one hour and solve for the exponential expression. By taking the natural logarithm of both sides of the equation, we simplify the exponential component, making it possible to isolate and solve for the cooling rate constant "k".
Similarly, to find the time "t" since the body was alive, we need to take the natural logarithm of the equation describing the cooling process. This step allows us to deal with the equation's exponential portion effectively, providing a linear form that makes solving for "t" straightforward.
The use of natural logarithms is critical in contexts involving exponential processes, turning complex multiplicative problems into more manageable additive ones.

Calculation of Cooling Rate

Calculation of the cooling rate is a fundamental component of applying Newton's Law of Cooling to real-world problems. The cooling rate "k" indicates how quickly the temperature of an object changes relative to the ambient temperature.
To determine "k" in this problem, we develop an equation by substituting known values of the body's temperature at two consecutive times. By simplifying and solving this equation through algebraic methods and natural logarithms, we calculate the rate at which the body cools. Having determined "k", we can estimate other unknowns such as the time of death. Calculating "k" helps predict the future or past temperatures of a cooling body by providing insights into how rapidly the temperature approaches that of the surrounding environment.
Understanding this rate is essential not only for solving theoretical problems but also in practical scenarios such as forensic investigations, where precise estimation of cooling rates can provide critical information about the timing of events.