Problem 61 - Calculus and its applications

Question

Find an expression relating the exponential growth rate \(k\) and the quadrupling time \(T_{4}\).

Step-by-Step Solution

Answer

The exponential growth rate \( k \) is \( k = \frac{\ln(4)}{T_4} \).

Step 1: Understand Exponential Growth

Exponential growth can be described by the function \( y(t) = y_0 e^{kt} \), where \( y_0 \) is the initial amount, \( k \) is the growth rate, and \( t \) is time.

Step 2: Define Quadrupling Time

The quadrupling time \( T_4 \) is the time it takes for the initial amount \( y_0 \) to quadruple. This means \( y(T_4) = 4y_0 \).

Step 3: Set Up the Equation for Quadrupling

Using the exponential growth function, set up the equation: \( 4y_0 = y_0 e^{kT_4} \).

Step 4: Simplify the Equation

Divide both sides of the equation by \( y_0 \): \( 4 = e^{kT_4} \).

Step 5: Take the Natural Logarithm

Take the natural logarithm of both sides to solve for \( kT_4 \): \( \ln(4) = kT_4 \).

Step 6: Solve for the Growth Rate \( k \)

Rearrange the equation to express \( k \) in terms of \( T_4 \): \( k = \frac{\ln(4)}{T_4} \).

Key Concepts

Growth Rate

Exponential growth is a fundamental concept often seen in real-world scenarios like population growth, finance, and biology. The growth rate, denoted by the variable \( k \), tells us how quickly the amount is increasing over time. It is a constant that appears in the exponential growth function, which is typically written as \( y(t) = y_0 e^{kt} \). This formula considers:

• \( y(t) \): the amount at time \( t \)
• \( y_0 \): the initial amount
• \( e \): the base of the natural logarithm (approximately 2.718)
• \( k \): the growth rate
• \( t \): time

The growth rate is crucial, as it gives insight into how fast the quantity is multiplying. A higher \( k \) means an increased speed of growth, indicating rapid exponential expansion. Understanding the growth rate allows us to predict future quantities by calculating how much an amount will grow over a specified period.

Quadrupling Time

Quadrupling time, denoted as \( T_4 \), defines how long it takes for a quantity to increase fourfold through exponential growth. In context, if you start with an initial amount \( y_0 \), quadrupling time is when your amount becomes \( 4y_0 \). The concept of quadrupling time helps us grasp how quickly growth is happening without having to monitor it continuously. Let's explore this through the exponential growth equation:Starting with the equation \( 4y_0 = y_0 e^{kT_4} \), we isolate what happens at quadrupling:

• Dividing both sides by \( y_0 \) results in \( 4 = e^{kT_4} \)
• This equation solidifies the definition: the time variable \( T_4 \) that makes the quantity four times the initial value

In various applications, knowing the quadrupling time can be crucial for strategic planning, like predicting population increases or anticipating financial investments.

Natural Logarithm

The natural logarithm, often symbolized as \( \ln \), is integral to solving equations in exponential growth. The natural logarithm has a unique base \( e \), which is a mathematical constant approximately equal to 2.718. It is commonly used in natural growth processes due to its intrinsic properties.In our context of quadrupling time, we solved \( 4 = e^{kT_4} \) by taking the natural logarithm of both sides:

• The result is \( \ln(4) = kT_4 \)
• This step transforms the problem into a manageable linear equation

Using the natural logarithm is crucial because it offers a straightforward method to "undo" the exponential aspect in the equation, simplifying our task of finding the growth rate \( k \). The relationship between exponential functions and their logarithms makes understanding one vital in solving for the other.