Problem 11
Question
The population of the United States can be modeled by the function \(\mathrm{p}(x)=80.21 e^{0.131 x}\) where \(x\) is the number of decades (ten year periods) since 1900 and \(\mathrm{p}(x)\) is the population in millions. a. Graph \(\mathrm{p}(x)\) over the interval \(0 \leq x \leq 15 .\) b. If the population of the United States continues to grow at this rate, predict the population in the years 2010 and \(2020 .\)
Step-by-Step Solution
Verified Answer
Graph the function for the interval \(0 \leq x \leq 15\). In 2010, the population is approximately 339 million, and in 2020 it is approximately 386.4 million.
1Step 1: Understanding the Function
The function for the population is given as \( \mathrm{p}(x) = 80.21 e^{0.131 x} \). This is an exponential function where \( x \) is the number of decades since 1900, and \( \mathrm{p}(x) \) gives the population in millions.
2Step 2: Setting the Interval for Graphing
We need to graph \( \mathrm{p}(x) \) for the interval \( 0 \leq x \leq 15 \). This means we start at the year 1900 (\( x = 0 \)) and go up to 2050 (\( x = 15 \)), as 15 decades past 1900 is 2050.
3Step 3: Plotting the Graph
To plot the graph, we evaluate \( \mathrm{p}(x) \) at several points within the interval. For example:- \( x = 0 \), \( \mathrm{p}(0) = 80.21 \times e^{0} = 80.21 \)- \( x = 5 \), \( \mathrm{p}(5) = 80.21 \times e^{0.655} \)- \( x = 10 \), \( \mathrm{p}(10) = 80.21 \times e^{1.31} \)- \( x = 15 \), \( \mathrm{p}(15) = 80.21 \times e^{1.965} \)Use these points to draw the curve, and it should exhibit an exponential growth pattern as \( x \) increases.
4Step 4: Calculating Population for 2010
To find the population in 2010, calculate \( \mathrm{p}(11) \) since 2010 is 11 decades after 1900. This gives:\[ \mathrm{p}(11) = 80.21 \times e^{0.131 \times 11} = 80.21 \times e^{1.441} \]Using a calculator, \( \mathrm{p}(11) \approx 80.21 \times 4.226 \approx 339.0 \) million.
5Step 5: Calculating Population for 2020
To find the population in 2020, calculate \( \mathrm{p}(12) \) as 2020 is 12 decades after 1900. This gives:\[ \mathrm{p}(12) = 80.21 \times e^{0.131 \times 12} = 80.21 \times e^{1.572} \]Using a calculator, \( \mathrm{p}(12) \approx 80.21 \times 4.819 \approx 386.4 \) million.
Key Concepts
Population ModelingExponential FunctionGraphing Exponential Functions
Population Modeling
Population modeling is a vital concept in understanding how populations grow over time. It uses mathematical functions to represent the growth patterns of a population, helping to predict future numbers. These models consider various factors such as birth rates, death rates, and migration.
For instance, in the provided exercise, we have a population model for the United States based on an exponential function. This model predicts the population based on the number of decades (\(x\)) since the year 1900.
The function used, \(\mathrm{p}(x) = 80.21 e^{0.131 x}\), estimates the population of the United States in millions. By using a specific mathematical approach, we can forecast the population size at different times, which aids in planning and policy-making. Such models are crucial for understanding future trends and making informed decisions.
For instance, in the provided exercise, we have a population model for the United States based on an exponential function. This model predicts the population based on the number of decades (\(x\)) since the year 1900.
The function used, \(\mathrm{p}(x) = 80.21 e^{0.131 x}\), estimates the population of the United States in millions. By using a specific mathematical approach, we can forecast the population size at different times, which aids in planning and policy-making. Such models are crucial for understanding future trends and making informed decisions.
Exponential Function
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It is represented by the formula \(f(x) = a e^{rx}\), where \(a\) is the initial amount, \(e\) is the base of natural logarithms (approximately 2.718), and \(r\) is the growth rate.
In our given function \(\mathrm{p}(x) = 80.21 e^{0.131 x}\), \(80.21\) is the initial population in millions, and \(0.131\) is the growth rate per decade. The role of the exponential function here is to show how the population increases over time, not by merely adding a fixed number but by growing at a rate proportional to its current size.
Exponential functions are essential in many fields, such as finance for interest calculations and biology for population growth, as they model processes that undergo continuous and accelerating change.
In our given function \(\mathrm{p}(x) = 80.21 e^{0.131 x}\), \(80.21\) is the initial population in millions, and \(0.131\) is the growth rate per decade. The role of the exponential function here is to show how the population increases over time, not by merely adding a fixed number but by growing at a rate proportional to its current size.
Exponential functions are essential in many fields, such as finance for interest calculations and biology for population growth, as they model processes that undergo continuous and accelerating change.
Graphing Exponential Functions
Graphing exponential functions helps visualize their growth patterns over time, making it easier to understand their characteristics. These graphs typically show a curve that starts slowly, increases rapidly, and becomes steeper over time.
In the exercise, we graph \(\mathrm{p}(x) = 80.21 e^{0.131 x}\) over the interval \(0 \leq x \leq 15\). This involves plotting points calculated at different values of \(x\), such as at \(x = 0, 5, 10,\) and \(15\), using the exponential equation. These points help draw the exponential curve, which visually demonstrates the acceleration in population growth.
When graphing, it's crucial to observe how changes in \(x\) affect \(\mathrm{p}(x)\) and the overall shape of the function. Most importantly, exponential graphs do not increase linearly; they exhibit continuous and rapidly increasing growth as seen in the steep ascent of the curve as \(x\) increases.
In the exercise, we graph \(\mathrm{p}(x) = 80.21 e^{0.131 x}\) over the interval \(0 \leq x \leq 15\). This involves plotting points calculated at different values of \(x\), such as at \(x = 0, 5, 10,\) and \(15\), using the exponential equation. These points help draw the exponential curve, which visually demonstrates the acceleration in population growth.
When graphing, it's crucial to observe how changes in \(x\) affect \(\mathrm{p}(x)\) and the overall shape of the function. Most importantly, exponential graphs do not increase linearly; they exhibit continuous and rapidly increasing growth as seen in the steep ascent of the curve as \(x\) increases.
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