Problem 415 - 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 \(145^{\circ} \mathrm{F}\) . Write a formula that models this situation.

Step-by-Step Solution

Answer

The formula is \(T(t) = 75 + 90e^{-0.405t}\).

Step 1: Understand the scenario and identify the key details

We are given that a turkey is taken out of the oven with an initial temperature of \(165^{\circ}F\), and it cools down to \(145^{\circ}F\) after 30 minutes in a \(75^{\circ}F\) room. We need to model this cooling process.

Step 2: Use Newton’s Law of Cooling

Newton's Law of Cooling tells us that the rate of temperature change of an object is proportional to the difference between the object's temperature and the ambient temperature. Mathematically, it can be expressed as:\[T(t) = T_r + (T_0 - T_r)e^{-kt}\]Where \(T(t)\) is the temperature at time \(t\), \(T_0\) is the initial temperature, \(T_r\) is the room temperature, and \(k\) is a constant.

Step 3: Substitute known values into the formula

From the problem, \(T_0 = 165^{\circ}F\), \(T_r = 75^{\circ}F\), and after 30 minutes (0.5 hours), \(T(0.5) = 145^{\circ}F\). Setting these into the equation gives:\[145 = 75 + (165 - 75)e^{-0.5k}\]

Step 4: Solve for the constant k

Simplify the equation:\[145 = 75 + 90e^{-0.5k}\]Subtract 75 from both sides:\[70 = 90e^{-0.5k}\]Divide by 90:\[\frac{70}{90} = e^{-0.5k}\]Take the natural logarithm of both sides to solve for \(k\):\[-0.5k = \, ln\left(\frac{70}{90}\right)\]Solve for \(k\):\[\k = -2 \times ln\left(\frac{70}{90}\right)\approx 0.405\]

Step 5: Write the complete formula

Now that we have \(k\), substitute it back into the Newton's Cooling formula:\[T(t) = 75 + (165 - 75)e^{-0.405t}\]This formula models the cooling of the turkey over time.

Key Concepts

Temperature Change

Temperature change in the context of Newton's Law of Cooling is all about how the temperature of an object decreases over time as it comes in contact with a cooler environment. The turkey, fresh from the oven, starts at a high temperature of 165°F. As it sits in a room at 75°F, it begins to cool down. This cooling process is not linear and doesn’t happen at a constant rate. Therefore, to understand this cooling behavior, we use mathematical models.

Temperature change is influenced by two key factors:

• The difference between the initial temperature of the object and the ambient temperature of the room. In our turkey example, this initial difference is 165°F - 75°F = 90°F.
• The rate at which the temperature decreases, which is affected by how long the object is exposed to the cooler temperature.

The underlying principle is that the greater the initial temperature difference, the faster the object will cool at the beginning. However, as the object approaches the ambient temperature, the rate of temperature change slows down.

Exponential Decay Model

The exponential decay model is a powerful mathematical tool used in Newton's Law of Cooling to describe how an object's temperature decreases over time. This model tells us that the change in temperature is not a constant drop straight to room temperature, but rather a curved decline that slows as it nears equilibrium.

In our turkey scenario, the temperature decrease follows this pattern:

• The model is represented by the formula: \(T(t) = T_r + (T_0 - T_r)e^{-kt}\), where \(T(t)\) represents the temperature at time \(t\), \(T_0\) is the initial temperature, \(T_r\) is the room temperature, and \(k\) is a constant for the cooling rate.
• This formula uses the exponential function \(e^{-kt}\), which causes the temperature to drop quickly at first and then slowly as it approaches room temperature \(T_r\).

Ultimately, this model helps to predict the turkey's temperature over time as it cools. The exponential model ensures we align with the real-world observation that cooling happens faster at the beginning when there's a larger temperature difference.

Constant of Proportionality

The constant of proportionality, often represented by \(k\), is a crucial component in Newton's Law of Cooling. It determines the rate at which an object's temperature approaches the ambient temperature. This value is unique to each situation; different environments and materials will have different constants.

To find \(k\), we use specific temperature data at a known time point. In our turkey problem, the temperature drops from 165°F to 145°F in 30 minutes. By placing these values into our cooling formula, we can solve for \(k\).

Here is the process briefly outlined:

• Start with the formula: \(145 = 75 + 90e^{-0.5k}\).
• Solve for \(k\) using algebra, yielding \(k \approx 0.405\).

This constant tells us how quickly the turkey cools in that specific room environment. A higher \(k\) would mean the turkey cools more rapidly, while a lower \(k\) signifies slower cooling. Knowing \(k\) allows us to predict future temperatures under similar conditions.