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Chapter 6: Problem 12

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The population of Clark County, Nevada, was 1.9 million in 2008. Assuming an annual growth rate of 4.5\%/yr, what will the county population be in \(2020 ?\)

Short Answer

Expert verified
Answer: Approximately 3,017,000.

Step by step solution

01

Identify the initial population and growth rate

The initial population, \(P_0\), is 1.9 million in the year 2008. The annual growth rate, \(r\), is 4.5\%.
02

Convert the annual growth rate to a decimal

To convert the annual growth rate into a decimal, divide by 100. Therefore, \(r = \frac{4.5}{100} = 0.045\).
03

Set up the exponential growth formula

The exponential growth formula is given by \(P(t) = P_0 e^{rt}\). With the given values, the formula becomes: \(P(t) = 1.9 \cdot 10^6 e^{0.045t}\), where \(t\) represents the number of years since 2008.
04

Calculate the population in 2020 using the exponential growth formula

To calculate the population in 2020, we need to find the value of \(t\) for 2020. Since the reference point \((t=0)\) is 2008, \(t = 2020 - 2008 = 12\) years. Now substitute this value of \(t\) into the formula: \(P(12) = 1.9 \cdot 10^6 e^{0.045 \cdot 12}\)
05

Evaluate the expression and interpret the result

Use a calculator or software to evaluate the expression: \(P(12) \approx 1.9 \cdot 10^6 e^{0.54} \approx 3.017 \cdot 10^6\) The population of Clark County, Nevada, in the year 2020 will be approximately 3,017,000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Growth
Population growth explores how a population changes over time. It's a concept that's fundamental when dealing with predictions in demography, urban studies, and environmental science.
For Clark County, understanding how the population has grown from 2008 to 2020 requires using mathematical models. These models help in knowing what resources might be needed in the future, like schools, roads, or hospitals.
Population growth can be influenced by many factors, such as
  • Birth rates and death rates
  • Migration
  • Economic conditions
  • Health care improvements
When designated rates like 4.5% for Clark County are provided, these show how quickly or slowly the population is expected to grow each year. Knowing this rate helps make more accurate predictions about population numbers in the future, which can then inform policies and planning.
Growth Rate
The growth rate is an essential piece in calculating population predictions. It tells you the speed at which a population increases every year.
The annual growth rate is often expressed as a percentage, like the 4.5% for Clark County. It's important to convert this percentage into a decimal for mathematical equations. So, 4.5% becomes a straightforward 0.045 by dividing by 100.
This rate is pivotal in transformative equations like the exponential growth formula. Misunderstanding growth rates can lead to enormous errors in calculations, because even small changes can have big impacts over many years.
Making estimations without this insight can lead policymakers, businesses, and community planners to make incorrect assumptions about future needs. Thus, understanding growth rates helps maintain accuracy in planning and response strategies.
Exponential Function
The exponential function is a powerful mathematical tool used to model growth. It's instrumental in various fields, particularly when predicting changes in populations.
In the case of exponential growth, the exponential function assumes the quantity grows by a fixed percentage over equal time periods, such as every year. For Clark County, the exponential growth formula used is:
  • \(P(t) = P_0 e^{rt}\)
Here, \(P_0\) is the initial population, and \(r\) is the growth rate as a decimal. The \(t\) is the time span in years since the reference point. \(e\) is a mathematical constant approximately equal to 2.71828.
The formula exponentially scales the population because with each passing year, the base population also adds to the growth, leading to a larger increase year by year. This is different from linear growth, where the amount added doesn't change.
Exponential functions can become quite complex as they adjust to real-world data, but they remain a key factor in making future estimates about populations and resources.

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