Problem 4 - Precalculus Essentials

Question

According to the National Center for Health Statistics, in \(1990,28 \%\) of babies in the United States were born to parents who were not married. Throughout the 1990 s, this percentage increased by approximately 0.6 per year. a. Express the percentage of babies born out of wedlock, \(P\), as a function of the number of years after \(1990, x\) b. If this trend continued, in which year were \(40 \%\) of babies born out of wedlock?

Step-by-Step Solution

Answer

a) The percentage of babies born out of wedlock, \(P\), as a function of the number of years after \(1990\), \(x\), is \(P(x) = 28 + 0.6x\). b) The year when \(40\%\) of babies were born out of wedlock is \(1990 +\) \(x = 1990 + \frac{40 - 28}{0.6} = \) approximately \(1990 + 20 = 2010\).

Step 1: Establish the Linear Function

Given that in \(1990=0\), \(28\%\) of babies were born out of wedlock and that this percentage increased by approximately \(0.6\%\) per year, this can be modeled by a linear function \(P(x)\), where \(x\) is number of years after \(1990\). Therefore, \(P(x) = 28 + 0.6x\).

Step 2: Solve for the Required Percentage

We want to calculate the year in which \(40\%\) of babies were born putting \(P(x) = 40\). We will now solve this equation for \(x\). So, \(40 = 28 + 0.6x\). Solving for \(x\) entails isolating \(x\) by subtracting \(28\) from both sides and then dividing by \(0.6\). Ergo, \(x = \frac{40 - 28}{0.6}\).

Step 3: Deduce the Year

Since \(x\) represents years after \(1990\), add \(x\) to \(1990\) to get the year when \(40\%\) of babies were born out of wedlock. So the year will be \(1990 + x\).

Key Concepts

Percentage Increase

Understanding the concept of percentage increase is essential when analyzing how values change over time. Consider a percentage as a way of expressing a proportion or fraction of 100. When something increases annually by a certain percentage, it means each year that the initial amount grows by that fixed percentage.
In the given problem, the percentage of babies born to unmarried parents increases by 0.6% per year. This is a consistent, steady increase, meaning the growth is linear over the years. So for instance:

• If the percentage was 28% in 1990, then in 1991 it will be 28.6%.
• In 1992, it will become 29.2%, and so on.

We can summarize this with a simple formula for linear growth, which is crucial when we express change over time.

Years Calculation

Calculating the number of years it takes for a percentage to reach a certain value involves understanding how time interacts with growth rates.
In our example, we start in 1990, knowing that's our base year. If we need to determine when another percentage is reached, say 40%, we need to figure out how many years it takes for the yearly increases to sum up to that new value.
Here is how you can think about it:

• Know your starting point – 1990 with 28%.
• Understand your end goal – 40%.
• Determine the needed increase – 12% (from 28% to 40%).

From there, estimate how many years it will take, given an increase of 0.6% each year, which involves some basic arithmetic and algebra.

Solving Linear Equations

To find out when a specific percentage of unmarried parents is reached, you must solve a linear equation. A linear equation is an equation that makes a straight line when plotted on a graph.
In this problem, the equation provided is:

• \(P(x) = 28 + 0.6x\)

This shows the starting point of 28%, plus a yearly increase modeled as 0.6x.
To find when the percentage reaches 40%, set \(P(x) = 40\) and solve for \(x\) by isolating \(x\).
Here's how you can do it:

• Subtract 28 from both sides: \(40 - 28 = 0.6x\)
• Divide by 0.6 to find \(x\): \(x = \frac{12}{0.6}\)

The value of \(x\) you find tells you how many years after 1990 the percentage will be 40%. This systematic approach to solving for \(x\) is a fundamental algebraic skill.