Problem 33 - College Algebra Essentials

Question

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(\left.2 A_{0}\right)\) is given by \(t=\frac{\ln 2}{k}\)

Step-by-Step Solution

Answer

The time it takes for a population to double using the exponential growth model is \(t=\frac{\ln 2}{k}\).

Step 1: Define the Problem

The exponential growth model is defined as \(A=A_{0} e^{k t}\). For a population reaching a doubled size, this becomes \(2A_{0}=A_{0} e^{k t}\).

Step 2: Simplify the Equation

Eliminate the common terms in both sides, which is \(A_0\). This simplifies the equation to \(2= e^{k t}\).

Step 3: Change to Logarithmic Form

To make \(t\) the subject, change the equation to logarithmic form (since the natural logarithm, \(\ln\), is the inverse of the exponential function). This results in \(\ln 2 =k t\).

Step 4: Make t Subject

To isolate \(t\) on one side of the equation, divide both sides by \(k\), so we get \(t=\frac{\ln 2}{k}\).

Key Concepts

Understanding Population Doubling Time

Population doubling time is a concept used to describe the time it takes for a population to grow to twice its initial size. This is commonly used in demographics and biology to evaluate how quickly a population, such as bacteria or humans, is growing. The exponential growth model provides an effective way to calculate this. In formulaic terms, when a population doubles in size, the equation denotes this as moving from \( A_0 \) to \( 2A_0 \). The doubling time is crucial for predicting population trends and potential future impacts on resources and environments.Using the exponential growth model \( A = A_0 e^{kt} \), you can derive the formula for doubling time \( t = \frac{\ln 2}{k} \). This shows that the time taken to double depends inversely on the rate of growth \( k \). A higher growth rate results in a shorter doubling time, highlighting its importance in planning and resource management.

Exploring Natural Logarithm

The natural logarithm is a fundamental mathematical function that plays a crucial role in exponential growth models. Represented as \( \ln \), the natural logarithm is the inverse function of the exponential \( e^x \). It helps convert exponential expressions into manageable linear ones.When we encounter exponential models like \( A = A_0 e^{kt} \), we use the natural logarithm to simplify and solve for variables. In our exercise, after simplifying the equation to \( 2 = e^{kt} \), taking the natural logarithm of both sides yields \( \ln 2 = kt \).The natural logarithm of a number \( y \) gives the exponent to which \( e \) must be raised to get \( y \). Thus, \( \ln 2 \approx 0.693 \), which is a consistent constant used in calculations involving exponential growth, especially for determining doubling time.

Unpacking Logarithmic Equations

Logarithmic equations like \( t = \frac{\ln 2}{k} \) are the result of simplifications in exponential problems. Converting exponential equations into a logarithmic form is a method used to extract variables, notably when the variable is situated exponently, like in \( e^{kt} \).To solve the exponential \( 2 = e^{kt} \), apply the natural logarithm to both sides, transforming it into a linear equation \( \ln 2 = kt \). Rewriting exponential equations as logarithmic ones allows for straightforward manipulation of variables, especially for isolating them.In terms of handling exponential growth problems, understanding logarithmic equations is vital. Mastering this:

• Allows for solving for unknowns.
• Turns multiplicative processes into additive ones, simplifying calculations.
• Provides insights into growth patterns and dynamics, crucial for strategic planning.

By learning how to tackle these equations, problems linked to growth or decay can be more easily managed and understood.