Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of \(165^{\circ} \mathrm{F}\) and is allowed to cool in a \(75^{\circ} \mathrm{F}\) room. After half an hour, the internal temperature of the turkey is \(145^{\circ} \mathrm{F}\). To the nearest degree, what will the temperature be after 50 minutes?

Step-by-Step Solution

Answer

The temperature will be approximately 124°F after 50 minutes.

Step 1: Understand the Problem

We have a turkey cooling from an initial temperature of \(165^{\circ} \mathrm{F}\) in a \(75^{\circ} \mathrm{F}\) room. After 30 minutes, the temperature is noted to be \(145^{\circ} \mathrm{F}\). We need to find the temperature after 50 minutes.

Step 2: Newton's Law of Cooling Formula

Newton's Law of Cooling is given by the formula: \[T(t) = T_r + (T_i - T_r) \, e^{-kt}\]where \(T(t)\) is the temperature at time \(t\), \(T_r\) is the room temperature, \(T_i\) is the initial temperature, and \(k\) is the cooling constant.

Step 3: Determine Constants and Plug Values

Here, \(T_r = 75^{\circ} \mathrm{F}\), \(T_i = 165^{\circ} \mathrm{F}\). At \(t = 30\) minutes, \(T(30) = 145^{\circ} \mathrm{F}\). Substitute these values into Newton's Law of Cooling to find \(k\).

Step 4: Calculate the Cooling Constant \(k\)

Using the values: \[145 = 75 + (165 - 75)e^{-30k}\]Subtract 75 from both sides:\[70 = 90e^{-30k}\]Divide both sides by 90:\[\frac{7}{9} = e^{-30k}\]Take the natural logarithm of both sides:\[\ln\left(\frac{7}{9}\right) = -30k\]Solve for \(k\):\[k = -\frac{1}{30}\ln\left(\frac{7}{9}\right)\]

Step 5: Calculate Temperature at 50 Minutes

Now, substitute \(k\) and solve for \(T(50)\):\[T(50) = 75 + (165 - 75)e^{-50k}\]Plug in the value of \(k\) calculated:\[T(50) = 75 + 90e^{-50(-\frac{1}{30}\ln\left(\frac{7}{9}\right))}\]Calculate \(e^{-50k}\) and find \(T(50)\).

Step 6: Simplify and Solve

Compute the exponent and the exponential function to obtain:\[e^{-50k} = \left(\frac{7}{9}\right)^{-\frac{5}{3}}\]Calculate the result and plug it back:\[T(50) \approx 75 + 90 \times 0.5412\]

Step 7: Final Calculation

Complete the multiplication and addition:\[T(50) \approx 75 + 48.71 \approx 124^{\circ} \mathrm{F}\]

Key Concepts

Exponential Decay

Exponential decay is a common concept in mathematics and physics that describes systems that decrease gradually over time. It is relevant in scenarios like cooling; hence, it is a crucial part of Newton's Law of Cooling. When we say "exponential decay," we refer to a process where the rate of decrease is proportional to the current value. This means as time passes, the value decreases at a rate that becomes slower over time. In our scenario, we observe how the turkey cools down in a room with a constant ambient temperature. This process is effectively represented by the exponential term in the formula:\[ T(t) = T_r + (T_i - T_r) e^{-kt} \]Here, the

• exponential term \(e^{-kt}\) dictates the decay rate
• negative sign inside the exponent ensures decay
• constant \(k\) determines how fast the process progresses

Understanding this process helps us predict how temperatures change over time, crucial in many technical and practical applications.

Temperature Modeling

Temperature modeling is a pivotal method employed in predicting how the temperature of an object changes relative to its environment. This specific role is played by Newton's Law of Cooling, which is a handy model for scientists and engineers. It utilizes the relationship between the internal temperature of an object and the surrounding ambient temperature to forecast temperature changes over time.In the exercise concerning the turkey, the model starts with initial conditions:

• the turkey's initial temperature \(T_i = 165^{\circ} \mathrm{F}\)
• the ambient room temperature \(T_r = 75^{\circ} \mathrm{F}\)

These parameters are then plugged into the cooling formula to solve for specific time intervals.The beauty of temperature modeling lies in its ability to simulate real-world cooling behaviors using analytical equations. This allows for precise predictions of when certain temperature thresholds are met, useful in cooking, chemical processes, and environmental studies. Learning this model empowers one to solve practical temperature-related challenges efficiently.

Problem-Solving

Effective problem-solving in real-world scenarios often demands an understanding of mathematical models like Newton's Law of Cooling. When faced with the task of determining the turkey’s temperature after 50 minutes, students must navigate through several analytical steps.First, it's crucial to understand the formulas involved, as illustrated:\[ T(t) = T_r + (T_i - T_r) e^{-kt} \]This equation is the foundation for problem-solving in cooling problems, and by inserting known values and computing unknowns (like the rate constant \(k\)), one can solve these types of questions.Critical steps in this process include:

• identifying known values like initial temperatures and times
• doing calculations to find constants or rates
• using algebraic manipulation and logarithms to solve for unknowns

Problem-solving not only demands mathematical calculative skills but also a structured approach to using those skills effectively in applied contexts, ensuring accuracy and deeper understanding in academics and beyond.