Problem 81 - Beginning and Intermediate Algebra

Question

One example of a mathematical model is an equation that describes the relationship between two quantities. For example, the life expectancy at birth of Americans can be approximated by the equation $$ y=0.157 x-237 $$ where \(x\) is a year between 1990 and 2015 and \(y\) is age in years. (Data from Centers for Disease Control and Prevention.) Use this model to approximate life expectancy (to the nearest year) in each of the following years. 1990

Step-by-Step Solution

Answer

75 years

Step 1: Identify the given information

From the problem, the year (x) is 1990. The equation given is: \( y = 0.157x - 237 \). This equation will be used to find the life expectancy (y) for the given year.

Step 2: Substitute the given year into the equation

Substitute \( x = 1990 \) into the equation: \( y = 0.157 \times 1990 - 237 \).

Step 3: Calculate the value inside the equation

Perform the multiplication first: \( 0.157 \times 1990 = 312.43 \).

Step 4: Complete the equation

Subtract 237 from the product obtained: \( 312.43 - 237 = 75.43 \).

Step 5: Round the result

Round the result to the nearest year: 75.43 rounds to 75.

Key Concepts

linear equations

Linear equations are fundamental in understanding relationships between variables. A linear equation is a mathematical statement that represents a straight line when plotted on a graph. It has the general form of:
\(y = mx + b\). Here:

• \(y\) is the dependent variable.


• \(x\) is the independent variable.


• \(m\) is the slope of the line.


• \(b\) is the y-intercept, or where the line crosses the y-axis.

In our exercise, the equation used is: y = 0.157x - 237. We see this follows the general form, with 0.157 being the slope (m), and -237 being the y-intercept (b).
Linear equations make it easy to predict one variable if we know the other. For instance, by substituting specific values of x, we can determine the corresponding values of y. This is especially useful in various fields, including economics, biology, and everyday life scenarios such as budgeting or calculating distances.
Understanding how to manipulate and utilize linear equations is a core skill in algebra, serving as a foundation for more complex mathematical concepts.

life expectancy calculation

Life expectancy calculations use linear equations to estimate how long people are expected to live based on certain factors such as the year of birth. The general formula given for this problem is: \(y = 0.157x - 237\), where x is a specific year and y represents the life expectancy in that year.
To find the life expectancy for a particular year, we simply plug in the value of that year into the equation and solve for y.

• Identify the year given in the problem (here, x = 1990).


• Substitute the year into the equation: \(y = 0.157 \times 1990 - 237\).


• Perform the multiplication: \(0.157 \times 1990 = 312.43\).


• Subtract the y-intercept: \(312.43 - 237 = 75.43\).


• Round to the nearest whole number: 75.43 rounds to 75.

This simple process shows how mathematical models can estimate meaningful statistics such as life expectancy, illustrating the practical application of linear equations in interpreting real-world data. This helps in planning and resource allocation in public health and many other sectors.

yearly data analysis

Yearly data analysis involves examining data collected over several years to identify trends, patterns, and insights. This can be crucial in many areas such as economics, healthcare, and environmental studies.
By analyzing yearly data, we can forecast future trends, evaluate past policies, and make informed decisions. In our exercise, the yearly data is used to analyze life expectancy from 1990 to 2015. The linear equation \(y = 0.157x - 237\) helps us interpret the life expectancy for any given year within this range.
Steps in Yearly Data Analysis:

• Gather and organize the data for each year.


• Use statistical tools and models (like linear equations) to identify trends.


• Interpret the results to understand the underlying patterns.


• Present the data using graphs and charts for better visualization.


• Make decisions based on the analysis.

In our example, substituting particular years into the equation gives us a clear picture of how life expectancy has changed over time. Understanding yearly data analysis ensures that we can create and validate mathematical models that provide critical insights and aid in forecasting future scenarios.