Question

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and ${45}^{\circ }\mathrm{F}$ outside and the temperature of the body is ${78}^{\circ }\mathrm{F}$. You know the cooling constant is $k={0.00824}^{\circ }\mathrm{F}/\min .$ When did the victim die, assuming that a human's temperature is ${98}^{\circ }\mathrm{F}$ ?

Short answer

The victim died at approximately 11:02 AM.

Step-by-step solution

Understand the Problem

We are given that the outside temperature is ${45}^{\circ }\mathrm{F}$ and the body's current temperature is ${78}^{\circ }\mathrm{F}$. The normal human body temperature is ${98}^{\circ }\mathrm{F}$, and we need to determine the time since death using the cooling constant $k={0.00824}^{\circ }\mathrm{F}/\min$.

Apply Newton's Law of Cooling Equation

Newton's Law of Cooling is given by the formula:$$T(t)=T_e+(T_0-T_e)\cdot e^{-kt}$$where $T(t)$ is the temperature at time $t$, $T_e$ is the ambient temperature, $T_0$ is the initial body temperature, and $k$ is the cooling constant.

Set Known Values in the Equation

In this case, we have: - $T(t)={78}^{\circ }\mathrm{F}$- $T_0={98}^{\circ }\mathrm{F}$- $T_e={45}^{\circ }\mathrm{F}$- $k={0.00824}^{\circ }\mathrm{F}/\min$Substitute these into the equation:$$78=45+(98-45)\cdot e^{-0.00824t}$$

Solve the Exponential Equation

First, subtract 45 from each side to isolate the exponential term:$$33=53\cdot e^{-0.00824t}$$Divide both sides by 53:$$e^{-0.00824t}=\frac{33}{53}$$Take the natural logarithm of both sides:$$-0.00824t=\ln \left(\frac{33}{53}\right)$$Calculate $t$:$$t=\frac{\ln \left(\frac{33}{53}\right)}{-0.00824}$$

Calculate Time Since Death

Perform the calculation:$$\ln \left(\frac{33}{53}\right)\approx -0.474$$Thus,$$t\approx \frac{-0.474}{-0.00824}\approx 57.52$$So, the time since death is approximately 57.52 minutes.

Determine Time of Death

Since it is currently noon, and approximately 57.52 minutes have passed since the victim died, the time of death would be roughly 11:02 AM.

Key concepts

Time of Death Estimation

Establishing the time of death is a crucial aspect of solving criminal cases. When a body is discovered, understanding how long it has been since the person passed away helps in reconstructing the events leading up to the death. This insight primarily relies on the known physical phenomena such as body cooling.

One method used by investigators is the application of Newton's Law of Cooling. In simple terms, this law helps you estimate how long it took for a body to cool from normal living temperatures to the temperature at the time of discovery. By using the initial temperature (in this case, the normal human body temperature of 98°F) and comparing it to the ambient temperature (here, 45°F), one can begin calculations.

• Ambient temperature (the temperature of the environment)
• Initial body temperature at the time of death
• Measured body temperature at the time of discovery

All these factors are utilized alongside a cooling constant to pinpoint the time elapsed since death.

Temperature Decay

Temperature decay refers to how the temperature of an object gradually aligns with that of its environment. This can be likened to how a hot cup of coffee gradually cools until it reaches room temperature. For human bodies, this decay process is instrumental in forensic fields to understand the time since death.

The decay is calculated using Newton's Law of Cooling, expressed as:
$$T(t)=T_e+(T_0-T_e)\cdot e^{-kt}$$
This formula considers several variables:

• $T(t)$: Current temperature of the body
• $T_e$: Ambient temperature
• $T_0$: Temperature when the organism was alive (initial)
• $k$: The cooling constant
• $t$: Time since the temperature began to drop

By rearranging this formula to solve for $t$, investigators can derive how long it has been since death occurred. Recognizing the decay pattern is vital for accurate computations in forensic science, as it provides a mathematical way to track heat loss.

Cooling Constant

The cooling constant, denoted by $k$, is a crucial parameter in Newton's Law of Cooling. This constant varies depending on several conditions, such as ambient temperature and the body's specific characteristics, influencing the rate at which the body temperature decreases.

In forensic analysis, understanding the specific cooling constant helps make more accurate estimations. It acts as a measure of how quickly the thermal energy leaves the body.

In our specific exercise, we have $k=0.00824\text{ degrees F/min}$. This means the body's temperature is decreasing at a very specific rate under the current conditions.

• A smaller $k$ value indicates slower cooling.
• A larger $k$ value implies a swifter temperature drop.

By integrating this constant into Newton's equation, forensic scientists can effectively calculate the time of death and provide answers in criminal investigations, allowing for a deeper understanding of the timeline of events.