Problem 25 - Precalculus with Calculus Previews

Question

A dead body was found within a closed room of a house where the temperature was a constant \(70^{\circ} \mathrm{F}\). At the time of discovery, the core temperature of the body was determined to be \(85^{\circ} \mathrm{F}\). One hour later a second measurement showed that the core temperature of the body was \(80^{\circ} \mathrm{F}\). Assume that the time of death corresponds to \(t=0\) and that the core temperature at that time was \(98.6^{\circ} \mathrm{F}\). Determine how many hours elapsed before the body was found.

Step-by-Step Solution

Answer

3 hours elapsed before the body was found.

Step 1: Understand Newton's Law of Cooling

Newton's Law of Cooling describes the rate of change of the temperature of an object to be proportional to the difference between its temperature and the ambient temperature. The formula is given by: \[ T(t) = T_a + (T_0 - T_a) e^{-kt} \] 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, and \( k \) is a constant.

Step 2: Identify Given Values

From the problem statement, we identify the following values:- Ambient temperature, \( T_a = 70^{\circ} \mathrm{F} \).- Initial temperature of the body, \( T_0 = 98.6^{\circ} \mathrm{F} \).- Temperature at discovery, \( T(\Delta t) = 85^{\circ} \mathrm{F} \).- Temperature after one hour, \( T(\Delta t + 1) = 80^{\circ} \mathrm{F} \).

Step 3: Set Up Equation for Time of Discovery

Using the formula from Step 1, we set the equation for the temperature \( 85^{\circ} \mathrm{F} \) when the body was discovered (time \( \Delta t \)):\[ 85 = 70 + (98.6 - 70) e^{-k\Delta t} \].

Step 4: Set Up Equation for One Hour Later

Now, we set up the equation for the temperature \( 80^{\circ} \mathrm{F} \) observed one hour after discovery (time \( \Delta t + 1 \)):\[ 80 = 70 + (98.6 - 70) e^{-k(\Delta t + 1)} \].

Step 5: Divide Equations to Remove Exponent

Divide the equation in Step 4 by the equation in Step 3 to eliminate the exponential term:\[ \frac{80 - 70}{85 - 70} = \frac{e^{-k(\Delta t + 1)}}{e^{-k\Delta t}} \].This simplifies to:\[ \frac{10}{15} = e^{-k} \].

Step 6: Solve for Decay Constant k

Take the natural logarithm of both sides to solve for \( k \):\[ \ln\left(\frac{2}{3}\right) = -k \].Thus, \( k = -\ln\left(\frac{2}{3}\right) \).

Step 7: Calculate Time Elapsed (Δt)

Substitute \( k \) into the equation for the body temperature at discovery:\[ 85 = 70 + (98.6 - 70) e^{-k\Delta t} \].Simplify to find \( \Delta t \):\[ \Delta t = -\frac{\ln\left(\frac{85-70}{98.6-70}\right)}{k} \].Substitute the value of \( k \) from Step 6 and solve for \( \Delta t \).

Step 8: Calculate Final Answer

After substituting all known values and simplifying the expression, the calculation reveals:\( \Delta t \approx 3 \) hours.This tells us that approximately 3 hours elapsed from the time of death until the body was discovered.

Key Concepts

Temperature Decay

Temperature decay refers to the process through which the temperature of an object decreases towards the ambient temperature. This process is largely driven by the exchange of heat between the object and its surroundings. In our scenario, the core temperature of a dead body was initially higher than its surrounding environment, causing it to cool down gradually. According to Newton's Law, the rate of temperature decay slows down as the object's temperature approaches the ambient level.
This cooling behavior follows a predictable pattern that can be described mathematically, allowing us to calculate how much time elapses as the temperature approaches equilibrium.

Exponential Decay

Exponential decay is a type of mathematical function describing processes that decrease at a rate proportional to their current value. In terms of Newton's Law of Cooling, the temperature of the body decreases exponentially over time, meaning it cools quickly at first, and then the rate of cooling slows.
The exponential component in the equation \( T(t) = T_a + (T_0 - T_a) e^{-kt} \) governs this behavior:

• \( T(t) \) indicates temperature at time \( t \).
• The factor \( e^{-kt} \) shows how the temperature approaches the ambient temperature over time.

Understanding exponential decay helps in analyzing how quickly the object moves towards thermal equilibrium.

Ambient Temperature

Ambient temperature is the consistent temperature of the surrounding environment within which the object is placed. In the given exercise, the ambient temperature is constant at \(70^{\circ} \mathrm{F}\). This value is crucial because the difference between the ambient temperature and the initial temperature of the body influences how the body temperature decays.

• The larger the difference between the initial and ambient temperatures, the faster the decay.
• The body will eventually stabilize at the ambient temperature if left undisturbed.

Understanding the role of ambient temperature helps to contextualize why calculations were set up as they were in the problem's solution, since it acts as the baseline towards which the body's temperature trends.

Thermal Equilibrium

Thermal equilibrium is the state in which the temperature of an object becomes equal to its surrounding environment, thus eliminating any temperature difference. In the context of the exercise, the body will achieve thermal equilibrium once it reaches the ambient temperature of \(70^{\circ} \mathrm{F}\).
This concept is important because it defines the endpoint of the cooling process and provides a reference point for understanding how the temperature changes over time.

• During this state, no net heat exchange occurs between the body and the environment.
• Thermal equilibrium is a stable condition—no further changes are observed under constant conditions.

Understanding thermal equilibrium helps to comprehend the ultimate outcome of the cooling process as dictated by Newton's Law.