Problem 43 - College Algebra

Question

Use this scenario: A turkey is taken out of the oven with an internal temperature of 165° Fahrenheit and is allowed to cool in a 75° F room. After half an hour, the internal temperature of the turkey is 145° F. Write a formula that models this situation.

Step-by-Step Solution

Answer

The model is \( T(t) = 75 + 90 e^{-(2 \ln(\frac{7}{9}))t} \).

Step 1: Understand the Cooling Process

The cooling of the turkey can be modeled using Newton's Law of Cooling. This law is expressed in the formula \( T(t) = T_r + (T_0 - T_r) e^{-kt} \), where \( T(t) \) is the temperature of the object at time \( t \), \( T_r \) is the room temperature, \( T_0 \) is the initial temperature of the object, and \( k \) is the cooling constant.

Step 2: Identify Known Values

From the given scenario, the initial temperature, \( T_0 \), of the turkey is 165°F, the room temperature, \( T_r \), is 75°F, and after 0.5 hours (30 minutes), the turkey's temperature \( T(0.5) \) is 145°F.

Step 3: Substitute Known Values into Formula

Start by substituting the known values into the Newton's Law of Cooling formula: \( 145 = 75 + (165 - 75) e^{-0.5k} \). This simplifies the equation to \( 145 = 75 + 90 e^{-0.5k} \).

Step 4: Solve for the Cooling Constant \( k \)

Rearrange the equation from the previous step to isolate the exponential term: \( 70 = 90 e^{-0.5k} \). Then divide both sides by 90: \( \frac{70}{90} = e^{-0.5k} \). Simplify to \( \frac{7}{9} = e^{-0.5k} \).

Step 5: Apply Natural Logarithm

Take the natural logarithm of both sides to solve for \( k \): \( ext{ln}\left(rac{7}{9}\right) = -0.5k \). Solve for \( k \) to get \( k = -2 ext{ln} \left(rac{7}{9}\right) \).

Step 6: Formulate the Final Model

Now, substitute the value of \( k \) back into the Newton's Law of Cooling formula. The temperature of the turkey at time \( t \) is given by: \( T(t) = 75 + 90 imes e^{-(2 ext{ln} \left(\frac{7}{9}\right))t} \), simplifying this using precise values for \( k \).

Key Concepts

Cooling Constant

The cooling constant, denoted as \( k \) in Newton's Law of Cooling, plays a crucial role in determining how fast an object cools in a given environment. It's an intrinsic property that defines the rate at which temperature decreases over time. In a broader sense, it's unique to each situation because it's influenced by factors like material composition, surface area, and specific heat capacity.
To find \( k \), you insert the given temperature values into the cooling formula and rearrange the equation. In our turkey scenario, the known values allowed us to isolate \( k \) to understand the temperature decline's specific character. This expression setup means any alterations in our initial conditions will impact \( k \), leading to diverse cooling behaviors.

• \( k \) determines cooling speed.
• A larger \( k \) means faster cooling.
• It's influenced by environmental and material factors.
  

Temperature Modeling

Temperature modeling using Newton's Law of Cooling enables us to predict an object's temperature over time accurately. This predictive model is valuable in various fields, from culinary applications to engineering and environmental science.
In our scenario, the model starts with the formula \[ T(t) = T_r + (T_0 - T_r) e^{-kt} \], where each component symbolizes a pivotal aspect of the cooling process. Here:

• \( T(t) \) is the temperature at time \( t \).
• \( T_r \) represents room temperature.
• \( T_0 \) is the object's initial temperature.

Using this model, we can calculate the turkey's temperature at any desired future time point by substituting the relevant values.
This approach provides a dynamic picture of the temperature change, offering insights into the turkey's readiness or safe storage potential.

Exponential Decay

The concept of exponential decay in thermal dynamics describes how temperature changes at a rate proportional to its current value, resulting in a rapid decline initially that slows over time. This mathematical phenomenon is evident in the equation \( e^{-kt} \) from our cooling model, where \( e \) denotes the base of natural logarithms.
This exponential function helps explain why the initial temperature drop from 165°F to 145°F in our turkey example is brisk, but the subsequent cooling process can take much longer to reach equilibrium or match ambient room temperature. The rate of change decreases as the temperature difference between the object and its environment shrinks.

• Exponentials model the rate of temperature decline.
• They create curves that level out over time.
• Exponential decay highlights initial rapid changes that taper off.

Thermal Dynamics

Thermal dynamics encompasses the principles governing heat transfer and temperature changes in objects, crucial in understanding processes like cooling. In our scenario, it highlights the interplay between the turkey's own heat and ambient conditions in the room.
The concepts of conduction, convection, and radiation could also influence the cooling process, but Newton's Law simplifies these complex interactions to a manageable model, considering the direct reduction of temperature difference.
Furthermore, understanding thermal dynamics not only helps in temperature modeling but also in planning around it, such as determining how long a cooked turkey should rest before serving for optimal taste or safety.

• Thermal dynamics examines heat transfer processes.
• It includes conduction, convection, and radiation.
• Concepts simplify with models like Newton's Law.

Thus, delving into thermal dynamics deepens our comprehension of cooling, aiding practical applications like safe food handling and energy conservation strategies.