Problem 58 - College Algebra with Modeling and Visualization

Question

Birth Rate In 1990 the number of births per 1000 people in the United States was 16.7 and decreasing at 0.21 birth per 1000 people each year. (Source: National Center for Health Statistics.) (a) Write a formula for a linear function \(f\) that models the birth rate \(x\) years after 1990 . (b) Estimate the birth rate in 2003 and compare the estimate to the actual value of 14 (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The birth rate function is \( f(x) = -0.21x + 16.7 \). In 2003, the estimated rate is 13.97, close to the actual 14.

Step 1: Understand the Problem

We are given that in 1990, the birth rate was 16.7 per 1000 people and that it decreases by 0.21 births per year per 1000 people. We need to find a linear function that models the birth rate over time.

Step 2: Determine the Function Parameters

To write a linear function model, we can use the formula for a linear function, which is \( f(x) = mx + b \). Here, \( b \) is the initial value (birth rate in 1990), which is 16.7, and \( m \) is the rate of change (decrease per year), which is \(-0.21\).

Step 3: Write the Linear Function

Substituting the values for \( m \) and \( b \) into the linear function formula, we get: \( f(x) = -0.21x + 16.7 \). This equation represents the birth rate \( x \) years after 1990.

Step 4: Calculate Birth Rate in 2003

The year 2003 corresponds to \( x = 13 \) because it is 13 years after 1990. Substitute \( x = 13 \) into the function: \[ f(13) = -0.21(13) + 16.7 \] \[ f(13) = -2.73 + 16.7 \] \[ f(13) = 13.97 \] Thus, the estimated birth rate in 2003 is 13.97 per 1000 people.

Step 5: Compare to Actual Value

The estimated birth rate in 2003 is 13.97. The actual recorded birth rate in 2003 was 14 per 1000 people. Our estimate closely matches the actual value, with a slight underestimate of 0.03.

Key Concepts

Linear Functions

A linear function is a mathematical function that creates a straight line when graphed. It's a simplified way to model a wide variety of relationships between two variables.
Linear functions are expressed in the form of the equation \( f(x) = mx + b \). This equation tells us that for every unit change in \( x \), \( f(x) \) changes by a constant amount, represented by the variable \( m \).
The simplicity of linear functions makes them particularly useful for modeling situations like the birth rate problem, where there's a consistent rate of change over time.

Rate of Change

The rate of change is a crucial concept in understanding linear functions. It is the 'slope' of the line and it determines how much the dependent variable changes with respect to a change in the independent variable.
In mathematical terms, it is represented by \( m \) in the linear function equation \( f(x) = mx + b \). For example, in the birthrate model, the rate of change \( m \) is \(-0.21\), which means for each year after 1990, the birth rate decreases by 0.21 per 1000 people.
This decrease is consistent each year, which is a characteristic feature of a linear relationship.

Function Parameters

Function parameters in the context of linear functions are the constants \( m \) and \( b \) in the equation \( f(x) = mx + b \). These parameters define the nature of the function.

• \( m \) is the slope or rate of change. It dictates the steepness of the line and the direction of change (increase or decrease).
• \( b \) is the y-intercept. It represents the initial value of the function when \( x = 0 \), which in this context, refers to the birth rate at the start year, 1990, given as 16.7 per 1000 people.

By substituting these parameters into the linear function formula, we can easily construct the mathematical model, which signifies relationships, like predicting the birth rate in future years using our derived equation \( f(x) = -0.21x + 16.7 \).