Problem 79 - Contemporary Precalculus

Question

The number of unmarried couples in the United States who live together was 3.2 million in 1990 and grew in a linear fashion to 5.5 million in 2000 . (a) Let \(x=0\) correspond to \(1990 .\) Write a linear equation expressing the number \(y\) of unmarried couples living together (in millions) in year \(x\). (b) Assuming the equation remains accurate, estimate the number of unmarried couples living together in 2010 . (c) When will the number of unmarried couples living together reach \(10,100,000 ?\)

Step-by-Step Solution

Answer

Answer: The estimated number of unmarried couples living together in 2010 is 7.8 million. The number of unmarried couples living together reaches 10,100,000 in the year 2020.

Step 1: Determine the linear equation

We are given that the number of unmarried couples was 3.2 million in 1990 (x=0) and increased to 5.5 million in 2000 (x=10). We can use these two points to determine the slope (m) of the linear equation using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In our case, \((x_1, y_1) = (0, 3.2)\) and \((x_2, y_2) = (10, 5.5)\). So: \(m = \frac{5.5 - 3.2}{10 - 0} = \frac{2.3}{10} = 0.23\) Now that we have the slope, we can write the linear equation in the form \(y = mx + b\). Since the number of unmarried couples was 3.2 million in 1990, when x = 0, we have: \(b = 3.2\) So the linear equation representing the number of unmarried couples, in millions, living together in year x is: \(y = 0.23x + 3.2\)

Step 2: Estimate the number of unmarried couples in 2010

To estimate the number of unmarried couples living together in 2010, we need to substitute x = 20 in the linear equation: \(y = 0.23(20) + 3.2 = 4.6 + 3.2 = 7.8\) So, the estimated number of unmarried couples living together in 2010 is 7.8 million.

Step 3: Find the year when the number of unmarried couples reaches 10,100,000

Since we need the year when the number of unmarried couples living together reaches 10.1 million, we will set y = 10.1 and solve for x: \(10.1 = 0.23x + 3.2\) Now, subtract 3.2 from both sides of the equation: \(6.9 = 0.23x\) To find x, divide both sides by 0.23: \(x = \frac{6.9}{0.23} \approx 30\) Since x = 0 corresponds to the year 1990, when x = 30, it corresponds to the year 1990 + 30 = 2020. Therefore, the number of unmarried couples living together will reach 10,100,000 in the year 2020.

Key Concepts

Linear Equation Problem Solving

Understanding how to create and solve linear equations is fundamental in precalculus. It allows students to model and predict real-world phenomena, such as trends in demography or economics. In this particular exercise, the problem required creating a linear equation to represent the growth in the number of unmarried couples living together.

First, we establish a relationship between two variables: the years since 1990 (\( x \)) and the number of unmarried couples living together (in millions, denoted as \( y \)). Since we're given two data points, these can be used as coordinates to find the slope (\( m \)) of the line, which represents the rate of change in the number of couples. After finding the slope, we use one of the points to solve for \( b \), the y-intercept, giving us the starting value in 1990 when \( x = 0 \). The resulting linear equation is \( y = 0.23x + 3.2 \).

Solving linear equations involves substituting known values into the equation to find the unknown variable—whether that's the number of couples in a future year or identifying the year when the number of couples hits a specific milestone. In this scenario, we predict the number for 2010 and calculate when the number will reach 10.1 million.

Slope Calculation

The slope is a critical concept in understanding linear equations. It tells us how much the dependent variable (in our case, the number of unmarried couples) changes per unit of the independent variable (the years since 1990).

To calculate the slope, or the rate of change, we use the formula \( m = \frac{\text{\textDelta } y}{\text{\textDelta } x} \), where \( \text{\textDelta } y \) is the change in the number of couples and \( \text{\textDelta } x \) is the change in years. The calculated slope of 0.23 indicates that for each year, the number of unmarried couples living together is increasing by 0.23 million, or 230,000 couples—an important statistic that helps us comprehend the pace and scale of societal changes.

Knowing how to calculate the slope enables us to create an equation and make predictions, making it an essential tool in problem solving.

Linear Growth Estimation

In problems dealing with linear models, understanding how to estimate future values is as vital as solving for the present. Linear growth estimation involves projecting current trends into the future, which sometimes can be linear, like in the case of our exercise.

Using the linear equation we've derived, \( y = 0.23x + 3.2 \), we can forecast the number of unmarried couples living together at any point in the future, so long as the trend remains linear. By substituting the number of years since 1990 (\( x \)) into the equation, we estimate the number of couples. For 2010, this was calculated to be 7.8 million. Such clear predictions can be pivotal in policy making, economics, or resource allocation in a community.

The linear model also allows us to work backward to figure out when certain thresholds will be crossed, as we did to determine that the number of unmarried couples living together is projected to reach 10.1 million in the year 2020. This extrapolation and prediction capability is a powerful application of precalculus to real-life situations.