Problem 37 - College Algebra

Question

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2003 to \(1,000,000\) people in \(2006 .\) Without showing the details, describeSuppose that a population that is growing exponentially increases from \(800,000\) people in 2003 to \(1,000,000\) people in \(2006 .\) Without showing the details, describe how to obtain the exponential growth function that models the data. how to obtain the exponential growth function that models the data.

Step-by-Step Solution

Answer

The exponential growth function that models the data is \(P(t) = 800,000 \times e^{(\frac{ln(1.25)}{3}) \times t}\).

Step 1: Understand the Exponential Growth function

The general form of an exponential growth function is \(P(t)=P_0 \times e^{rt}\), where: \n - \(P(t)\) is the population at time \(t\), - \(P_0\) is the initial population size, - \(e\) is the base of natural logarithms, approximately equal to 2.71828, - \(r\) is the growth rate, and - \(t\) is the time.

Step 2: Substitute the given data into the equation

Given are two points on the curve defined by the population growth: \(P_0 = 800,000\) (the population in 2003) and \(P(3) = 1,000,000\) (the population in 2006; this is 3 because 2006 is three years after 2003). Plug these values into the exponential growth equation: \[1,000,000 = 800,000 \times e^{3 \times r}\].

Step 3: Solve for the rate of growth \(r\)

This is a simple exponential equation for \(r\). First, divide both sides of the equation by 800,000 to isolate the exponential term on the right side: \[\frac{1,000,000}{800,000} = e^{3 \times r}\]. This simplifies to: \[1.25 = e^{3 \times r}\]. To solve for \(r\), take the natural logarithm (ln) of both sides: \[ln(1.25) = ln(e^{3 \times r})\]. The property of logarithms states that \(ln(e^{x}) = x\), so the right side simplifies to \(3 \times r\): \[ln(1.25) = 3 \times r\]. Divide both sides by 3 to solve for \(r\): \[r = \frac{ln(1.25)}{3}\].

Step 4: Construct the exponential growth equation

Insert the calculated value of \(r\) from Step 3 back into the general form of the exponential growth function to obtain the growth function that models this data: \(P(t) = 800,000 \times e^{(\frac{ln(1.25)}{3}) \times t}\).

Key Concepts

Understanding the Exponential Function

An exponential function is a mathematical expression used to describe situations of constant exponential growth or decay. It has the general form \( y = a \times e^{bx} \), where:

• \( y \) is the value of the function at time \( x \).
• \( a \) is the initial amount or starting value.
• \( e \) is the base of natural logarithms, approximately equal to 2.71828.
• \( b \) is the growth (or decay) rate constant.
• \( x \) represents time or another independent variable.

Exponential functions are powerful tools in describing phenomena that grow or shrink exponentially, such as populations, investments, or radioactive decay. They highlight how the growth rate depends on the current value, leading to faster growth as the value becomes larger.
To construct an exponential function, you need to determine the initial value and the growth rate. These parameters allow the function to model real-world exponential growth scenarios accurately.

Population Modeling with Exponential Functions

Population modeling is a crucial application of exponential functions, as it helps to predict how populations change over time. In exponential population modeling, the formula \( P(t) = P_0 \times e^{rt} \) is often used. Here:

• \( P(t) \) is the population at time \( t \).
• \( P_0 \) is the initial population size.
• \( r \) is the exponential growth rate.
• \( t \) is the amount of time that has passed.

This model is effective when population growth is proportional to its current size, meaning the larger the population, the faster it grows.
For example, if a town's population increases from 800,000 to 1,000,000 over three years, you can use specific values in the model to predict future population or understand past trends. It's an essential tool in planning, such as infrastructure development or resource management, given population implications.

Calculating the Growth Rate

Determining the growth rate \( r \) is critical in understanding how a population or quantity grows over time. To find \( r \) in an exponential growth scenario, follow these steps:Start with the equation \( 1,000,000 = 800,000 \times e^{3r} \) that comes from putting real-world data into the form \( P(t) = P_0 \times e^{rt} \).
First, solve for the exponential component by dividing on both sides: \( \frac{1,000,000}{800,000} = e^{3r} \). This simplifies to \( 1.25 = e^{3r} \).
Next, use the natural logarithm to eliminate the exponential: taking \( \ln(1.25) = \ln(e^{3r}) \). The natural logarithm exploits the property where \( \ln(e^x) = x \), simplifying to \( 3r = \ln(1.25) \).
Finally, find \( r \) by dividing both sides by 3: \( r = \frac{\ln(1.25)}{3} \). This shows how \( r \) quantifies the exponential growth independently of time or initial specific values.

Role of Natural Logarithm in Solving Exponential Equations

The natural logarithm, denoted \( \ln(x) \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828. It's a fundamental part of calculus and is especially useful in solving exponential equations.
In exponential growth modeling, natural logarithms help us 'unlock' growth rates from the exponent. For instance, when you have an equation like \( e^{3r} = 1.25 \), taking the natural logarithm of both sides allows you to bring down the exponent:
Using the property \( \ln(e^x) = x \), the right-hand side simplifies to \( 3r \). The left-hand side becomes \( \ln(1.25) \). So, the equation \( e^{3r} = 1.25 \) transforms into \( \ln(1.25) = 3r \).
This process of using natural logarithms is crucial because it converts the multiplicative relationships found in exponential equations into additive ones, which are much easier to solve. This makes natural logarithms a key player in handling exponential expressions in mathematics.