Problem 81 - College Algebra with Corequisite Support

Question

Newton's Law of Cooling states that the temperature \(T\) of an object at any time \(t\) can be described by the equation \(T=T_{s}+\left(T_{0}-T_{s}\right) e^{-k t},\) where \(T_{s}\) is the temperature of the surrounding environment, \(T_{0}\) is the initial temperature of the object, and \(k\) is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.

Step-by-Step Solution

Answer

\(t = -\frac{1}{k} \ln\left(\frac{T - T_s}{T_0 - T_s}\right)\).

Step 1: Isolate the Exponential Term

Start with the equation given by Newton's Law of Cooling: \[ T = T_s + (T_0 - T_s)e^{-kt} \]Subtract \( T_s \) from both sides to isolate the exponential term:\[ T - T_s = (T_0 - T_s)e^{-kt} \]

Step 2: Divide to Further Isolate the Exponential

Divide both sides by \((T_0 - T_s)\) to completely isolate the exponential component:\[ \frac{T - T_s}{T_0 - T_s} = e^{-kt} \]

Step 3: Apply the Natural Logarithm

Take the natural logarithm of both sides to solve for the exponent:\[ \ln\left(\frac{T - T_s}{T_0 - T_s}\right) = \ln\left(e^{-kt}\right) \]

Step 4: Use Logarithm Properties

Since \(\ln(e^x) = x\), simplify the right side:\[ \ln\left(\frac{T - T_s}{T_0 - T_s}\right) = -kt \]This uses the property that the natural logarithm and the exponential function are inverses.

Step 5: Solve for Time \(t\)

Rearrange the equation to solve for \(t\):\[ t = -\frac{1}{k} \ln\left(\frac{T - T_s}{T_0 - T_s}\right) \]This equation now expresses \(t\) as a single logarithm.

Key Concepts

Exponential Function

Newton's Law of Cooling is an excellent demonstration of exponential functions in action. An exponential function is a powerful mathematical concept that describes growth or decay based on a constant rate of change. In our case, this relates to how heat is lost over time. The equation given by Newton's Law of Cooling, \[ T = T_s + (T_0 - T_s) e^{-kt} \], involves an exponential decay, as it uses the Euler's number \( e \) raised to the power of \(-kt\). Here are key things to know:

• **Exponential Decay**: If the exponent is negative, such as \(-kt\), the function describes a decay process, meaning the quantity decreases over time.
• **Base of Exponential Function**: The base \(e\) is a special number approximately equal to 2.718. It frequently appears in nature, finance, and other fields, especially in continuous growth processes.
• **Rate of Change**: The parameter \(k\) represents the cooling rate, which affects how quickly the object's temperature approaches the surrounding temperature \(T_s\).

Understanding these aspects of exponential functions can help you visualize and predict how the temperature changes over time. Recognizing that the exponential component helps determine the rate of temperature loss will aid in solving for time in cooling problems.

Properties of Logarithms

Logarithms are the inverse operations of exponentials and are essential tools in solving exponential equations like Newton's Law of Cooling. In this exercise, logarithms are used to isolate the time variable \( t \). We use the natural logarithm because we are dealing with the exponential function with base \( e \). Here are several properties of logarithms that come in handy:

• **Logarithm of a Quotient**: \( \ln\left( \frac{A}{B} \right) = \ln(A) - \ln(B) \), useful when simplifying logarithmic expressions.
• **Logarithm of a Power**: \( \ln(A^b) = b \cdot \ln(A) \), allowing us to bring down exponents in logarithmic equations.
• **Inverse Property**: Since \( \ln(e^x) = x \), this property helps simplify and solve equations where the exponential function is present.

By understanding and applying these properties, you can rearrange and solve for \( t \) efficiently. The step-by-step solution demonstrates using the inverse property to transition from an exponential equation to a more straightforward linear form, making the problem solvable with basic algebra.

Natural Logarithm

A natural logarithm, denoted as \( \ln \), is a specific type of logarithm that uses Euler's number \( e \) as its base. In many natural processes, including cooling and growth, the natural logarithm is used because it relates to the continuous rate of change.

• **Euler's Number \( e \)**: This number is approximately 2.718 and is used in many mathematical and real-world applications.
• **Solving Exponential Equations**: The natural logarithm is especially useful for solving equations involving \( e \), since its primary property allows \( \ln(e^x) = x \). This property simplifies the process of finding variables such as time \( t \) when addressing problems like Newton's Law of Cooling.
• **Importance in Physics and Math**: Because the natural logarithm deals with rates of growth and decay, it is widely used in physics, mathematics, and other sciences to model and analyze natural patterns and behaviors.

Embracing the concept of the natural logarithm can aid in understanding the steps required to solve exponential equations efficiently, and highlight its versatility and power in various applications.