Problem 417 - Precalculus

Question

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of \(165^{\circ}\) o Fahrenheit 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 minute, how long will it take the turkey to cool to \(110^{\circ} \mathrm{F} ?\)

Step-by-Step Solution

Answer

It takes about 89 minutes for the turkey to cool to 110°F.

Step 1: Understand the Scenario

We are given that the initial temperature of the turkey is 165°F and it cools in a room with a constant temperature of 75°F. After 30 minutes, its temperature is 145°F. We need to find out how long it takes for the turkey to cool down to 110°F.

Step 2: Apply 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 temperature and the ambient temperature. The formula for this is given by:\[ T(t) = T_s + (T_0 - T_s) e^{-kt}\]where:- \(T(t)\) is the temperature of the object at time \(t\).- \(T_s\) is the surrounding temperature (75°F).- \(T_0\) is the initial temperature (165°F).- \(k\) is the cooling constant.- \(t\) is the time in minutes.

Step 3: Set Up Equation for Given Conditions

First, use the information after 30 minutes, when the temperature is 145°F to find \(k\). Substitute into the formula:\[145 = 75 + (165 - 75)e^{-30k}\]This simplifies to:\[145 = 75 + 90e^{-30k}\]\[70 = 90e^{-30k}\]\[e^{-30k} = \frac{70}{90}\]\[e^{-30k} = \frac{7}{9}\]

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

Take the natural logarithm of both sides to solve for \(k\):\[-30k = ext{ln}rac{7}{9}\]\[k = -rac{ ext{ln}rac{7}{9}}{30}\]

Step 5: Calculate Time to Reach 110°F

Use the cooling constant \(k\) in the equation to find when the temperature is 110°F:\[110 = 75 + (165 - 75)e^{-kt}\]\[35 = 90e^{-kt}\]\[e^{-kt} = \frac{35}{90}\]\[e^{-kt} = \frac{7}{18}\]Take the natural logarithm:\[-kt = ext{ln}rac{7}{18}\]Substitute \(k\) from Step 4:\[t = -rac{ ext{ln}rac{7}{18}}{k}\]Substitute \(k = -rac{ ext{ln}rac{7}{9}}{30}\) into the equation for \(t\):\[t = 30 \frac{ ext{ln}rac{7}{18}}{ ext{ln}rac{7}{9}}\]Calculate the value of \(t\).

Step 6: Provide Final Answer

Using the above calculations, determine \(t\). Use a calculator for the natural log expressions and perform the division to find that it takes approximately 89 minutes for the turkey to cool to 110°F.

Key Concepts

Temperature Change

Temperature change refers to how the temperature of an object, like a turkey taken out of the oven, fluctuates over time when exposed to an environment with a different temperature. In the exercise, the turkey begins at an internal temperature of 165°F and is left in a room with an ambient temperature of 75°F. Thus, the turkey begins to cool.
Temperature change follows a natural process, depending on several factors, such as:

• The initial temperature of the object, which is the starting point for the cooling process.
• The ambient temperature of the surrounding environment, which acts as a constant baseline temperature.

Understanding temperature change is crucial to predicting how long it will take for an object like the turkey to reach a specific temperature, such as when it should be served at 110°F.

Cooling Constant

The cooling constant is a significant parameter in Newton's Law of Cooling, which helps determine how quickly an object cools. It is denoted as "k" in the mathematical formula and is unique to every situation because it depends on factors like the material properties of the object and the nature of the surroundings.
To find the cooling constant, one needs to understand that more significant differences in temperature between the object and the surroundings generally lead to a faster cooling rate:

• Use data from a specific moment, such as the turkey's temperature after 30 minutes, to calculate "k" using the derived equation involving natural logarithms.
• The value of "k" is crucial for subsequent calculations to predict future temperature changes accurately.

By solving for "k," we can accurately predict how much longer it will take for the turkey to reach its desired temperature.

Ambient Temperature

Ambient temperature is the typical or natural temperature of the environment surrounding an object. It serves as a reference point in Newton's Law of Cooling, against which the temperature change of the object is measured.
In our exercise, the ambient temperature is 75°F, which is the temperature of the room where the turkey is left to cool. This constant temperature influences the rate at which the turkey's temperature decreases:

• It acts as a base level that the cooling process moves towards.
• The closer the object's temperature is to the ambient temperature, the slower the cooling rate becomes, following an exponential decay pattern.

Recognizing the role of ambient temperature is vital in predicting how an object adjusts its temperature over time.

Exponential Decay

Exponential decay is a mathematical concept that describes the process by which quantities decrease over time. In the context of Newton's Law of Cooling, it models the decrease in the temperature of an object like our turkey as it cools towards the ambient temperature.
Instead of a constant linear decline, the process follows a curved path where the rate of temperature decrease slows as the turkey approaches the room’s ambient temperature.
This can be expressed using the formula:\[T(t) = T_s + (T_0 - T_s) e^{-kt}\]Here, the terms "e^-kt" represent the exponential factor causing the temperature to drop over time.

• The curve demonstrates that rapid changes occur early on, then gradually reduce.
• This model helps us calculate specific time intervals needed to reach desired temperatures based on initial conditions.

Understanding exponential decay allows us to plot and predict the cooling process effectively.