Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Solve the differential equation for Newton's Law of Cooling for an arbitrary \(T_{0}, T_{1}\), and \(k\), assuming that \(T_{0}>T_{1}\). Show that \(\lim _{t \rightarrow \infty} T(t)=T_{1}\).

Step-by-Step Solution

Answer

The solution is \(T(t) = T_1 + (T_0 - T_1) e^{-kt}\), and \(\lim_{t \to \infty} T(t) = T_1\).

Step 1: Understanding Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation \[ \frac{dT}{dt} = -k(T - T_1) \], where \(T(t)\) is the temperature at time \(t\), \(T_1\) is the ambient temperature, and \(k\) is a positive constant. Our task is to solve this equation given that \(T_0 > T_1\).

Step 2: Setting up the Initial Condition

We know that at \(t = 0\), the temperature of the object is \(T_0\), that is, \(T(0) = T_0\). We will use this to find the specific solution to the differential equation.

Step 3: Solving the Differential Equation

This is a separable differential equation. We start by separating the variables: \[ \frac{dT}{T-T_1} = -k dt \]. Integrating both sides, we get \[ \int \frac{1}{T-T_1} dT = -k \int dt \]. This yields \[ \ln|T - T_1| = -kt + C \], where \(C\) is the integration constant.

Step 4: Solving for the Constant

Apply the initial condition \(T(0) = T_0\). Substitute into the equation: \[ \ln|T_0 - T_1| = C \]. Thus, the equation becomes \[ \ln|T - T_1| = -kt + \ln|T_0 - T_1| \].

Step 5: Expressing the Solution as a Function of Time

Exponentiate both sides to solve for \(T\): \[ |T - T_1| = e^{-kt} \, |T_0 - T_1| \]. This becomes \[ T - T_1 = \, (T_0 - T_1) \, e^{-kt} \] because \(T_0 > T_1\), so the absolute value sign is not needed. Hence, \[ T(t) = T_1 + (T_0 - T_1) e^{-kt} \].

Step 6: Finding the Limit as \(t\) Approaches Infinity

We need to evaluate \( \lim_{t \to \, \infty} T(t) \). Substitute \(T(t) = T_1 + (T_0 - T_1) e^{-kt} \). As \(t\) approaches infinity, \(e^{-kt}\) approaches zero. Thus, the limit is \( T_1 \).

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling provides a model for how an object's temperature changes over time when exposed to a surrounding environment. The key idea is that the rate of temperature change of the object is proportional to the difference between its own temperature and the ambient temperature. This relationship is expressed by the differential equation:

• \( \frac{dT}{dt} = -k(T - T_1) \)

Here, \(T(t)\) is the temperature of the object at time \(t\), \(T_1\) is the ambient temperature, and \(k\) is a positive constant that depends on the nature of the object and the environment. This equation tells us that if the object's temperature \(T\) is greater than \(T_1\), the temperature will decrease at a rate proportional to \(T - T_1\). Conversely, if \(T\) is less than \(T_1\), the object will warm up.

Separable Differential Equation

A separable differential equation is one in which the variables can be separated on different sides of the equation. This form simplifies solving the equation greatly. In the case of Newton's Law of Cooling, we have:

• \( \frac{dT}{dt} = -k(T - T_1) \)
• The goal is to isolate \(dT\) and \(dt\) on opposite sides: \( \frac{dT}{T-T_1} = -k \, dt \).

This equation is now separable because we can integrate both sides separately. Doing so provides the integrals:

• \( \int \frac{1}{T-T_1} \, dT = \int -k \, dt \)

The integral of the left side involves the natural logarithm function, \( \ln|T - T_1| \), while that of the right side yields \(-kt+C\), where \(C\) is an integration constant.

Temperature Function

After solving the separable differential equation, we arrive at the function that describes how temperature changes over time. From the logarithmic form, we have:

• \( \ln|T - T_1| = -kt + C \)

By exponentiating both sides and using the initial conditions, where \( T(0) = T_0 \), we deduce:

• \( T(t) = T_1 + (T_0 - T_1)e^{-kt} \)

This temperature function shows an exponential decay towards \(T_1\). As time progresses, the object's temperature either cools or warms, depending on whether it starts above or below the ambient temperature \(T_1\). The rate of this change is governed by the constant \(k\), which indicates how quickly equilibrium is reached.

Limit of a Function

The concept of a limit helps determine the behavior of a function as the input approaches some value. In `Newton's Law of Cooling`, we are interested in what happens to the temperature function \(T(t)\) as time \(t\) goes towards infinity.Looking at the temperature function:

• \( T(t) = T_1 + (T_0 - T_1)e^{-kt} \)

As \( t \to \infty \), the term \( e^{-kt} \) approaches zero because it represents exponential decay. Therefore, the remaining expression simplifies to:

• \( T(t) \to T_1 \)

This shows that over a long period, the temperature of the object will approach the ambient temperature \(T_1\). Intuitively, it means that regardless of the initial temperature \(T_0\), the object will eventually reach thermal equilibrium with its surroundings.