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FORENSIC SCIENCE The temperature \(T\) (in degrees Celsius) of the body of a murder victim found in a room where the air temperature is \(20^{\circ} \mathrm{C}\) is given by $$ T(t)=20+17 e^{-0.07 t} $$ where \(t\) is the number of hours after the victim's death. a. Graph the body temperature \(T(t)\) for \(t \geq 0\). What is the horizontal asymptote of this graph, and what does it represent? b. What is the temperature of the victim's body after 10 hours? How long does it take for the body's temperature to reach \(25^{\circ} \mathrm{C}\) ? c. Abel Baker is a clerk in the firm of Dewey, Cheatum, and Howe. He comes to work early one morning and finds the corpse of his boss, Will Cheatum, draped across his desk. He calls the police, and at 8 A.M., they determine that the temperature of the corpse is \(33^{\circ} \mathrm{C}\). Since the Last item entered on the victim's notepad was, "Fire that idiot, Baker," Abel is considered the prime suspect. Actually, Abel is bright cnough to have been reading this text in his spare time. He glances at the thermostat to confirm that the room temperature is \(20^{\circ} \mathrm{C}\). For what time will he need an alibi to establish his innocence?

Step by step solution

01

Understanding the Function

The temperature of the body over time is given by the function \[ T(t) = 20 + 17e^{-0.07t} \] where \( T \) is in degrees Celsius and \( t \) is in hours. Identify the constants and the variables.

02

Graphing the Function

Graph the function \( T(t) = 20 + 17e^{-0.07t} \) for \( t \geq 0 \) using a graphing tool or manually by calculating \( T(t) \) for different values of \( t \). Notice the curve's behavior as \( t \) increases.

03

Determine the Horizontal Asymptote

As \( t \) approaches infinity, the exponential term \( e^{-0.07t} \) approaches 0. Thus, \( T(t) \) approaches 20. The horizontal asymptote is \( T = 20 \), which represents the room temperature.

04

Calculate Temperature After 10 Hours

Substitute \( t = 10 \) into the function: \[ T(10) = 20 + 17e^{-0.07 \times 10} \] Calculate the value to find the temperature after 10 hours.

05

Find When Temperature Reaches 25°C

Set \( T(t) = 25 \) and solve for \( t \): \[ 25 = 20 + 17e^{-0.07t} \] Isolate the exponential term and solve for \( t \).

06

Determine Time of Death

Given \( T(0) = 33 \) at 8 AM, set up the equation \[ 33 = 20 + 17e^{-0.07t} \] to find the value of \( t \). Solve for \( t \) to determine how many hours before 8 AM the death occurred and ascertain the time of death.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay

Exponential Decay describes the process of reducing an amount by a consistent percentage rate over a period of time.
In forensic science, it is used to model the cooling of a body after death. The formula given in the problem is T(t) = 20 + 17e^{-0.07t} The term 17e^{-0.07t} represents the exponential decay. The base of the natural logarithm, e, raised to any power, shows how quickly the decreasing happens. In this case, the decay rate is 0.07 (or 7%) per hour. The exponential term decreases as t increases, meaning the temperature difference between the body and room temperature lessens over time.

Newton's Law of Cooling

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. In the given problem, the ambient temperature is 20°C, and the changing body temperature follows the formula given: T(t) = 20 + 17e^{-0.07t}. Here, the ambient temperature ( T_a = 20°C) is a critical factor. When the room temperature is reached, the cooling process slows considerably. The larger the initial temperature difference, the faster the initial rate of cooling.

Horizontal Asymptote

A horizontal asymptote in a graph represents a value that the function approaches but never actually reaches as the input approaches infinity. For the body temperature model {latex} T(t) = 20 + 17e^{-0.07t}, {latex} this value is {latex} T = 20. This model implies that as time goes on, regardless of the initial temperature of the body, it will eventually cool down to the room temperature of 20°C. The horizontal asymptote serves an important interpretive function. In this case, it tells us the long-term temperature equilibrium of the body with its surroundings.

Graphing Exponential Functions

When graphing exponential functions like {latex} T(t) = 20 + 17e^{-0.07t}, {latex} we need to observe certain features:

• Starting point: At time t = 0, the initial temperature is 20 + 17e0 = 37°C.
• Decay behavior: The exponent’s coefficient, -0.07, dictates how quickly the curve approaches the horizontal asymptote.
• Long-term behavior: As t increases, e^{-0.07t} approaches 0, causing T(t) to asymptotically approach 20°C. Creating a table of values for t (0, 1, 2, 3, …) and T(t) helps visualize this.

The graph confirms the decreasing trend of the temperature and its gradual approach toward 20°C.

Temperature-Time Relationship

The relationship between temperature and time here shows how body heat is lost after death according to a predictable model. Specifically, the temperature decrease follows an exponential decay pattern as described by {latex} T(t) = 20 + 17e^{-0.07t}. With time, the function indicates the temperature (T) will get closer and closer to the ambient room temperature but never go below it. On a forensic level, understanding this relationship helps estimate the time of death, which is crucial for criminal investigations. For example, solving for {latex} t when {latex} T(t) = 33°C, allows us to backtrack and estimate how long it has been since the victim's body began cooling.