Problem 107 - College Algebra and Calculus: An Applied Approach

Question

The population \(y\) (in millions of people) of North America from 1980 to 2050 can be modeled by \(y=5.3 x+377, \quad-20 \leq x \leq 50\) where \(x\) represents the year, with \(x=50\) corresponding to 2050\. (Source: U.S. Census Bureau) (a) Find the \(y\) -intercept of the graph of the model. What does it represent in the given situation? (b) Construct a table of values for \(x=-20,-10,0,10\), \(20,30,40\), and 50 (c) Plot the solution points given by the table in part (b) and use the points to sketch the graph of the model.

Step-by-Step Solution

Answer

The y-intercept of the graph of the model is 377, this represents the population (in millions) of North America in 1980. The table of values at x=-20,-10,0,10,20,30,40, and 50 are 367, 372, 377, 382, 387, 392, 397, and 402 respectively. Lastly, using these points, the graph is plotted to represent North America's projected population growth over the years.

Step 1: Find Intercept

The y-intercept of the graph is the point where the line intercepts the y-axis. This point occurs when \( x = 0 \). Therefore, by substituting \( x = 0 \) into the equation, we get: \( y = 5.3(0) + 377 = 377 \). Hence, the y-intercept of the graph is 377.

Step 2: Create Value Table

Creating a value table will help to visualize the relationship between x (the year) and y (population). We would replace \( x \) with the proposed years and calculate the respective 'y' (population). This leads to following values: for \( x = -20, y = 367 \); for \( x = -10, y = 372 \); for \( x = 0, y = 377 \); for \( x = 10, y = 382 \); for \( x = 20, y = 387 \); for \( x = 30, y = 392 \); for \( x = 40, y = 397 \); and finally for \( x = 50, y = 402 \).

Step 3: Draw Graph

Having obtained the value table, each pair (x,y) represents a point in the graph. These points are plotted on a graph, and a line is drawn through those points. The line represents the population growth trend in North America from 1980 to 2050.

Key Concepts

Population Modeling

Population modeling is a crucial aspect of understanding demographic changes over time. In this context, mathematical models are used to predict future population sizes based on current data and trends. The given model, \(y=5.3x+377\), is a linear function that represents the population of North America between 1980 and 2050. Here, \(x\) stands for the years since 1980, while \(y\) represents the population in millions.

Such models help policymakers and researchers to visualize and anticipate population growth or decline. They can be essential for planning resources, infrastructure, and services needed to accommodate future populations. By understanding the slope and intercept in the equation, we can identify the rate of population change and the population at a specific starting point.

Graphing Linear Equations

Graphing linear equations like \(y=5.3x+377\) allows us to visualize how the population is expected to change over time. In a graph, the x-axis typically represents time (in years in this case), and the y-axis represents the population. Each point on the line corresponds to a specific year and its predicted population.

By plotting the equation, we create a visual representation of the population model. The slope, 5.3, indicates how rapidly the population increases each year. A steeper slope would mean a faster rate of change. This sketch becomes a powerful tool in predicting and comparing the population at different times.

Y-intercept

The y-intercept is a fundamental concept in linear equations. It's where the graph crosses the y-axis, which happens when \(x = 0\). In this population model, substituting \(x = 0\) into the equation gives \(y = 377\).

The y-intercept, 377, represents the starting population in the context of the problem. This specific point refers to the population size in the base year, 1980, when \(x\) equates to 0. Understanding the y-intercept helps us anchor our predictions with a concrete reference point in time.

Table of Values

A table of values is a practical tool for visual learning and understanding linear relationships. In this exercise, creating a table involves selecting different values for \(x\) and computing their corresponding \(y\) values using the model equation \(y=5.3x+377\).

This tabulation clarifies how the population increases with each passing decade. It offers specific data points, such as: for \(x = -20\), \(y = 367\), and for \(x = 50\), \(y = 402\).

• Helps pinpoint certain years and their predictions
• Aids in plotting these points on a graph
• Gives numerical backing to the graphical representation

Creating a value table is an excellent exercise for validating the model and preparing to draw its graph.