Problem 6 - Algebra 2

Question

For Exercises 6 and \(7,\) use the following information. Fayette County, Kentucky, grew from a population of \(260,512\) in 2000 to a population of \(268,080\) in \(2005 .\) Write an exponential growth equation of the form \(y=a e^{k t}\) for Fayette County, where \(t\) is the number of years after 2000 .

Step-by-Step Solution

Answer

The exponential growth equation is \( y = 260,512 \cdot e^{0.0057t} \).

Step 1: Understand the Exponential Growth Model

The exponential growth model is defined as \( y = a e^{kt} \). Here, \( y \) is the population at time \( t \), \( a \) is the initial population, \( e \) is the base of natural logarithms, and \( k \) is the growth rate. Our task is to find \( k \) and write the function.

Step 2: Identify the Known Values

The initial population \( a \) in the year 2000 is given as \( 260,512 \). The population in 2005 is \( 268,080 \). The time \( t \) in years from 2000 to 2005 is \( 5 \).

Step 3: Set Up the Equation from 2005 Data

For the year 2005, substitute the known values into the equation: \( 268,080 = 260,512 \, e^{5k} \).

Step 4: Solve for the Growth Rate k

First, divide both sides of the equation by \( 260,512 \), resulting in \( \frac{268,080}{260,512} = e^{5k} \). Calculate the left side to get approximately \( 1.029 \). Next, take the natural logarithm of both sides to solve for \( k \), resulting in: \( 5k = \ln(1.029) \). Solve for \( k \) by dividing by \( 5 \) to get \( k \approx \frac{\ln(1.029)}{5} \approx 0.0057 \).

Step 5: Substitute k to Write the Equation

Finally, substitute \( a = 260,512 \) and \( k \approx 0.0057 \) back into the exponential growth model to get the equation \( y = 260,512 \, e^{0.0057t} \).

Key Concepts

Growth Rate

The growth rate in exponential growth scenarios refers to how quickly something is increasing over time.
In this case, it's the population of a locality, Fayette County, that's growing. Understanding growth rate is crucial in determining how fast the population changes from year to year.

In mathematical terms, this growth rate is represented by the variable \( k \) in the equation. A positive \( k \) suggests an increase, while negative indicates a decline.

• The step-by-step method to find \( k \) involves using known values from a certain time period.
• By dividing the final population by the initial population, you get a growth factor.

This factor helps in calculating \( k \) using natural logarithms.
By solving for \( k \), you determine the rate at which Fayette County's population grew between 2000 and 2005.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent.
Here, the base is Euler's number, \( e \), which is approximately 2.718, and the variable is \( t \), representing time.

Exponential functions are ideal for modeling situations like population growth where quantities increase over time. They are represented as \( y = a e^{kt} \), where:

• \( a \) indicates the starting value of the population.
• \( e \) is the base of the natural logarithm, ensuring a natural growth process.
• \( k \) is the growth rate we discussed earlier.

In the context of our exercise, finding the correct form of the exponential function involves deriving an equation that accurately reflects how the population evolves.
This equation can then predict future population sizes given different values of \( t \).

Population Growth

Population growth describes how the number of individuals in a specific area increases over a set period.
This concept is particularly relevant to regions experiencing either urban expansion or natural growth due to factors like birth rates.

Understanding population growth can have significant implications for resource management, city planning, and economic forecasts. Mathematically, we approach population growth using models like the exponential growth function.

• These models offer a snapshot of current dynamics, allowing predictions of future population sizes.
• In Fayette County, this growth was captured through data from the years 2000 to 2005.

By applying known data to the exponential model, we formed an equation that links past data with expectations.
This connection is essential for long-term planning, determining infrastructure needs, and fostering sustainable development.