Problem 22 - College Algebra

Question

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to triple (to grow from \(A_{0}\) to \(3 A_{0}\) ) is given by \(t=\frac{\ln 3}{k}\)Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population

Step-by-Step Solution

Answer

The time it takes a population to triple (to grow from \(A_{0}\) to \(3 A_{0}\)) using the exponential growth model is given by \(t=\frac{\ln 3}{k}\).

Step 1: Setting Up the Equation

First, substitute \(3A_{0}\) for \(A\) in the exponential growth model equation, to get the equation: \(3A_{0}=A_{0} e^{k t}\).

Step 2: Simplifying the Equation

Next, simplify the equation by dividing both sides by \(A_{0}\). This leaves us with the equation: \(3=e^{k t}\).

Step 3: Applying Natural Logarithm

Now, we will apply the natural logarithm to both sides to allow us to solve for \(t\). We'll get the equation: \(\ln 3 = k t\).

Step 4: Solving for t

Finally, we can solve for \(t\) by dividing both sides by \(k\). This gives us: \(t=\frac{\ln 3}{k}\).

Key Concepts

Population Growth

Understanding the dynamics of population growth is critical for a range of fields, from biology to economics. It describes how the size of a population changes over time and is often modeled with equations that reflect exponential trends. An exponential growth model, like the one described in the exercise, is one such equation.

The model can be simplified as follows: starting with a baseline population, denoted as A_0, the population size A at any later time t can be predicted by the exponential equation A = A_0 e^{kt}. Here, k represents the growth rate, and e is the base of the natural logarithm, which is approximately equal to 2.71828.
e is crucial as it reflects continuous growth, and the formula can be used to calculate the time needed for the population to reach any multiple of its original size. For instance, to find the time it takes for a population to triple, we adhere to the steps in the given solution and get an equation involving the natural logarithm of 3.

Natural Logarithm

The natural logarithm is an operation that is the inverse of exponentiation with the base e. In simpler terms, while exponentiation asks the question, 'to what power must we raise e to obtain a certain number?', the natural logarithm answers, 'Given a number, what power do we raise e to achieve it?'.

It's represented by the symbol ln, and you apply it to both sides of an equation involving an exponential term to simplify and solve for the variable of interest. In our case, ln is applied to both sides of the exponential equation to solve for time t. This process transforms the equation from its exponential form into a linear form that is much easier to handle algebraically.
The properties of natural logarithms are extensively used in solving problems related to growth and decay, as illustrated in the textbook exercise.

Algebraic Problem Solving

Algebraic problem solving involves identifying patterns, using abstraction, and manipulating algebraic expressions and equations to find solutions to real-world problems. The process of solving exponential growth models, like the one in our population growth problem, showcases a common algebraic technique: transforming an equation to make a variable solvable.

In the exemplified steps, we manipulated the initial exponential equation by dividing both sides to isolate the exponential-term, and then we applied the natural logarithm to both sides to ultimately solve for t. A systematic approach—setting up the equation, simplifying, applying appropriate algebraic operations (like logarithms), and isolating the desired variable—is the crux of algebraic problem solving.

It's important to practice and master these steps to tackle similar problems effectively, especially when they involve exponential terms and require the use of logarithms to unravel.