Problem 66 - A Graphical Approach to College Algebra

Question

Asian-American populations (in millions) are shown in the table. $$\begin{array}{|l|c|c|c|c|}\hline \text { Year } & 2003 & 2005 & 2007 & 2009 \\\\\hline \begin{array}{l}\text { Population } \\\\\text { (in millions) }\end{array} & 11.8 & 12.6 & 13.3 & 14.0\end{array}$$ (a) Use the points \((2003,11.8)\) and \((2009,14.0)\) to find the point-slope form of a line that models the data. \(\operatorname{Let}\left(x_{1}, y_{1}\right)=(2003,11.8)\) (b) Use this equation to estimate the Asian-American population in 2013 to the nearest tenth of a million.

Step-by-Step Solution

Answer

The estimated population in 2013 is 13.9 million.

Step 1: Find the Slope

First, we need to find the slope of the line. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the points \((2003, 11.8)\) and \((2009, 14.0)\) into the formula:\[m = \frac{14.0 - 11.8}{2009 - 2003} = \frac{2.2}{6} = 0.3667.\]

Step 2: Write the Point-Slope Form Equation

Using the point-slope form of the equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (2003, 11.8)\) and \(m = 0.3667\), we have:\[y - 11.8 = 0.3667(x - 2003).\]

Step 3: Convert to Slope-Intercept Form

Now, distribute the slope and simplify to get the equation in the slope-intercept form, \(y = mx + b\):\[y - 11.8 = 0.3667x - 734.9753.\]Adding 11.8 to both sides gives us:\[y = 0.3667x - 723.1753.\]

Step 4: Estimate the Population in 2013

Using the equation from Step 3, estimate the population in 2013 by substituting \(x = 2013\):\[y = 0.3667(2013) - 723.1753 = 737.1 - 723.1753 = 13.9.\]The Asian-American population in 2013 is approximately 13.9 million.

Key Concepts

Slope Calculation

To understand how to find the slope of a line, imagine connecting two points on a graph. The slope tells us how steep that line is.
It's like measuring how much one thing increases compared to another. For example, in our exercise, we have two points: \((2003, 11.8)\) and \((2009, 14.0)\). These points represent different years and their corresponding population sizes in millions.
To calculate the slope, we use the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• \(y_2 - y_1\) is the change in population, which is \(14.0 - 11.8\)
• \(x_2 - x_1\) is the change in years, which is \(2009 - 2003\)

So, the slope \(m\) is equal to \(\frac{2.2}{6} \approx 0.3667\).
Think of the slope as a measure of change: it shows the population increase per year.

Linear Equation

Once we have the slope, we use it to create a linear equation in the point-slope form. This equation models the relationship between the years and the population sizes.
The point-slope form of a linear equation is represented as:

• \(y - y_1 = m(x - x_1)\)
• Here, \((x_1, y_1)\) is one of our original points. For example, \( (2003, 11.8)\)
• And \(m\) is our calculated slope, \(0.3667\).

Thus, the equation becomes \(y - 11.8 = 0.3667(x - 2003)\).
This formula gives us a blueprint for finding the population at any given year.
So if we want to predict future populations, we simply plug in the desired year as \(x\).

Population Estimation

Using the linear equation, we can estimate future populations such as in 2013.
First, convert the point-slope form to a more straightforward format called slope-intercept form:

• Distribute the slope: \(0.3667(x - 2003)\).
• Combine terms to simplify: \(y - 11.8 = 0.3667x - 734.9753\).
• Add 11.8 to both sides to find \(y\): \(y = 0.3667x - 723.1753\).

This slope-intercept form \(y = mx + b\) makes it easy to estimate the population in any given year.
For 2013, plug \(x = 2013\) into the equation:

• \(y = 0.3667(2013) - 723.1753\).
• This computes to \(y \approx 13.9\).

Therefore, in 2013, the estimated Asian-American population is approximately 13.9 million.
This approach provides a simple method for projecting population trends based on past data.