Problem 24 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If a population is growing exponentially, then the time it takes the population to quadruple is independent of the size of the population.

Step-by-Step Solution

Answer

True, the quadrupling time is independent of the initial population size.

Step 1: Understanding Exponential Growth

Exponential growth implies that a population grows by a constant percentage rate over time. Mathematically, exponential growth can be expressed as \( P(t) = P_0 e^{rt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, and \( e \) is the base of the natural logarithm.

Step 2: Setting Up the Quadrupling Condition

To determine when the population quadruples, set \( P(t) = 4P_0 \). This means the population has grown to four times its initial size. Substitute into the exponential growth equation: \( 4P_0 = P_0 e^{rt} \).

Step 3: Simplifying the Equation

Divide both sides of the equation by \( P_0 \): \( 4 = e^{rt} \). This simplifies the problem to finding when the exponential term reaches 4.

Step 4: Solving for Time (t)

To solve for \( t \), take the natural logarithm of both sides: \( \ln(4) = rt \). Now, solve for \( t \): \( t = \frac{\ln(4)}{r} \).

Step 5: Analyzing the Result

Notice that the expression \( t = \frac{\ln(4)}{r} \) does not contain \( P_0 \). This indicates that the time it takes for the population to quadruple depends only on the growth rate \( r \) and not on the initial size \( P_0 \).

Key Concepts

Population Dynamics

Population dynamics refer to the changes and factors that affect how populations grow, shrink, or stabilize over time. When studying these dynamics, it's essential to understand the factors driving population changes, such as birth rates, death rates, and migration.

In the context of exponential growth, population dynamics are simplified. The growth rate, represented by a constant percentage increase, results in a population that continues to grow rapidly over time. Regardless of its initial size, a population experiencing exponential growth will maintain this pattern as long as it is subject to the same growth factors.

Understanding these dynamics is crucial in fields like ecology, economics, and epidemiology. It helps scientists and policymakers predict trends and make informed decisions.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena through mathematical expressions. In the context of population growth, mathematical models can predict future population sizes based on current data and trends.

The model for exponential growth is given as \( P(t) = P_0 e^{rt} \). This equation helps us model how populations evolve over time. The equation's components – initial population \( P_0 \), growth rate \( r \), and time \( t \) – are crucial for making accurate predictions.

By setting up mathematical models like this, we can solve various questions, such as determining how long it will take for a population to quadruple. This is essential for planning and resource allocation in fields like conservation and urban development.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is an essential mathematical function often used in equations involving exponential growth or decay. It helps us transform exponential equations into a more manageable form.

In solving problems related to population growth, the natural logarithm proves invaluable. For example, when you need to find out how long it will take for a population to quadruple, you start with the equation \( 4 = e^{rt} \).

By applying the natural logarithm, you simplify this to \( \ln(4) = rt \), allowing you to easily solve for time \( t \) by dividing \( \ln(4) \) by the growth rate \( r \).

This demonstrates the natural logarithm's utility in breaking down complex exponential growth problems, making them more approachable and easier to solve.