Problem 26 - Precalculus Mathematics for Calculus

Question

These exercises use Newton’s Law of Cooling. Newton's Law of Cooling is used in homicide investigations to determine the time of death. The normal body temperature is \(98.6^{\circ} \mathrm{F}\). Immediately following death, the body begins to cool. It has been determined experimentally that the constant in Newton's Law of Cooling is approximately \(k=0.1947,\) assuming that time is measured in hours. Suppose that the temperature of the surroundings is \(60^{\circ} \mathrm{F}\). (a) Find a function \(T(t)\) that models the temperature \(t\) hours after death. (b) If the temperature of the body is now \(72^{\circ} \mathrm{F}\), how long ago was the time of death?

Step-by-Step Solution

Answer

The function is \( T(t) = 60 + 38.6 e^{-0.1947t} \) and the death occurred approximately 7 hours ago.

Step 1: Understanding the Law

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

Step 2: Defining the Temperature Function

Given that \( T_s = 60^{\circ} \mathrm{F} \), \( T_0 = 98.6^{\circ} \mathrm{F} \), and \( k = 0.1947 \), we substitute these values into the formula:\[ T(t) = 60 + (98.6 - 60) e^{-0.1947t} \]Simplifying gives\[ T(t) = 60 + 38.6 e^{-0.1947t} \] This is the temperature function modeling the body's temperature \( t \) hours after death.

Step 3: Setting Up the Equation for Time of Death

We need to find \( t \) when the body temperature \( T(t) = 72^{\circ} \mathrm{F} \). Set the temperature function equal to 72 and solve for \( t \) :\[ 72 = 60 + 38.6 e^{-0.1947t} \]

Step 4: Isolating the Exponential Term

Subtract 60 from both sides to isolate the exponential expression:\[ 12 = 38.6 e^{-0.1947t} \] Then divide both sides by 38.6:\[ \frac{12}{38.6} = e^{-0.1947t} \]

Step 5: Solving for Time

Take the natural logarithm of both sides to solve for \( t \):\[ \ln\left(\frac{12}{38.6}\right) = -0.1947t \] Divide by \(-0.1947\) to solve for \( t \):\[ t = \frac{\ln\left(\frac{12}{38.6}\right)}{-0.1947} \] Calculating gives \( t \approx 7 \) hours.

Key Concepts

Temperature Modeling

Temperature modeling is the process of predicting how the temperature of an object changes over time. In this context, we're looking at a human body cooling after death.
This is important because it helps investigators estimate the time of death, which can be crucial in solving crimes.
Newton's Law of Cooling is a fundamental principle used in temperature modeling. It involves parameters like:

• Ambient temperature \(T_s\): the temperature of the environment around the object in question. In the exercise, it's given as \(60^\circ \text{F}\).
• Initial temperature \(T_0\): the temperature of the object at time zero, which is usually the normal body temperature, \(98.6^\circ \text{F}\).
• Time \(t\): the time elapsed since death.
• Cooling constant \(k\): a specific value that describes the rate at which the body cools. In this case, \(k = 0.1947\).

The formula used, \(T(t) = T_s + (T_0 - T_s) e^{-kt}\), models this cooling process mathematically.

Exponential Decay

Exponential decay is a process where the value of something decreases at a rate proportional to its current value. This principle is essential in understanding how the temperature of a cooling body changes over time.
In the context of Newton's Law of Cooling, the rate of temperature change is captured by an exponential decay function. Here's how it works:

• Start with an exponential component \(e^{-kt}\), where \(e\) is the exponential function, \(k\) is the cooling constant, and \(t\) is time. As time passes, \(t\) increases, making \(-kt\) more negative, leading the whole expression to decrease exponentially.
• This characteristic of exponential decay explains why the body temperature drops more slowly as it approaches the surrounding temperature.

It is a crucial part of solving the problem and helps determine how fast the temperature declines over time.

Homicide Investigation

Homicide investigation often involves determining the time of death, crucial for establishing timelines and possible suspects.
Newton's Law of Cooling provides a mathematic basis to calculate how much time has passed since death based on the body's current temperature. This is done by:

• Setting the known body temperature, 72°F in this exercise, equal to the temperature function \(T(t) = 60 + 38.6 e^{-0.1947t}\).
• Rearranging and solving this equation to find \(t\), which represents the time elapsed since death.
  This calculation, when executed correctly, gave an elapsed time of approximately 7 hours.

By accurately modeling and solving this equation, investigators can make educated estimates on the time of death, thus contributing vital information to a homicide investigation.