Problem 62
Question
Find a linear function in slope-intercept form that models the given description. Each function should model the percentage of total spending, \(p(x)\), by Americans \(x\) years after 1950 . In 1950 , Americans spent \(3 \%\) of their budget on health care. This has increased at an average rate of approximately \(0.22 \%\) per year since then.
Step-by-Step Solution
Verified Answer
The function that models the percentage of total spending by Americans x years after 1950 is \(p(x) = 0.22x + 3\).
1Step 1: Identify the slope
Given that the spending increased by 0.22% each year, this means that the slope (m) of the equation is 0.22.
2Step 2: Identify the y-intercept
The percentage of total spending in the base year, 1950, is given as 3%. This serves as the y-intercept (c) of this function.
3Step 3: Formulate the function in slope-intercept form
By substituting the slope (m) and y-intercept (c) into the slope-intercept form, the function that represents the percentage of total spending, \(p(x)\), by Americans x years after 1950 can be written as: \(p(x) = 0.22x + 3\).
Key Concepts
Slope-Intercept FormY-InterceptModeling with FunctionsRate of Change
Slope-Intercept Form
The slope-intercept form of a linear equation is a way of writing the equation of a line so that you can easily identify its slope and y-intercept. The formula is given by
This form is incredibly useful because you can quickly see how the line behaves just by looking at two numbers. The slope tells you how steep the line is and in which direction it goes, whereas the y-intercept tells you where the line crosses the y-axis, or in other words, when \( x = 0 \).
In our example, the slope-intercept form helps in constructing a function to describe how healthcare spending has changed over the years.
- \( y = mx + c \)
This form is incredibly useful because you can quickly see how the line behaves just by looking at two numbers. The slope tells you how steep the line is and in which direction it goes, whereas the y-intercept tells you where the line crosses the y-axis, or in other words, when \( x = 0 \).
In our example, the slope-intercept form helps in constructing a function to describe how healthcare spending has changed over the years.
Y-Intercept
The y-intercept of a linear function is the point where the line crosses the y-axis. It is represented by \( c \) in the slope-intercept form \( y = mx + c \).
This is the value of the function when \( x = 0 \). In the context of the exercise, the y-intercept \( c = 3 \) represents the percentage of total spending on healthcare in 1950.
Knowing the y-intercept is crucial for understanding the starting point of the function you are modeling. It allows you to interpret the function's initial value at the beginning of the time period in question.
Thus, the y-intercept gives us historical context in the exercise by marking the amount already being spent on healthcare before any changes occurred.
This is the value of the function when \( x = 0 \). In the context of the exercise, the y-intercept \( c = 3 \) represents the percentage of total spending on healthcare in 1950.
Knowing the y-intercept is crucial for understanding the starting point of the function you are modeling. It allows you to interpret the function's initial value at the beginning of the time period in question.
Thus, the y-intercept gives us historical context in the exercise by marking the amount already being spent on healthcare before any changes occurred.
Modeling with Functions
Modeling with functions is a powerful mathematical technique used to represent real-world situations with equations. By using functions, you can make predictions, understand trends, and analyze data.
In this exercise, modeling describes the consistent increase in healthcare spending over time.
Using a linear model allows us to simplify the complex changes into an easy-to-understand formula that shows continuous, consistent growth over time. This makes it digestible even for those who have just begun learning about linear functions.
In this exercise, modeling describes the consistent increase in healthcare spending over time.
- The function \( p(x) = 0.22x + 3 \) captures how spending grows each year.
Using a linear model allows us to simplify the complex changes into an easy-to-understand formula that shows continuous, consistent growth over time. This makes it digestible even for those who have just begun learning about linear functions.
Rate of Change
Rate of change in a function refers to how much the function's value changes as the variable changes. In a linear function, this rate of change is constant and is called the slope.
For the linear function we produced, the slope \( m = 0.22 \) is the rate of change. It shows that the percentage of healthcare spending increases by 0.22% every year.
In essence, the rate of change gives us a complete picture of the dynamics at play in a modeled situation.
For the linear function we produced, the slope \( m = 0.22 \) is the rate of change. It shows that the percentage of healthcare spending increases by 0.22% every year.
- This constant change highlights an average annual growth, providing insight into how rapidly or slowly a phenomenon evolves over time.
In essence, the rate of change gives us a complete picture of the dynamics at play in a modeled situation.
Other exercises in this chapter
Problem 61
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