Problem 34 - College Algebra

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 increased by approximately \(0.6 \%\) per year. Use this information to solve Exercises \(33-34\) If this trend continues, in which year will \(40 \%\) of babies be born out of wedlock?

Step-by-Step Solution

Answer

The year is \(20\) years after \(1990\), which is \(2010\)

Step 1: Understanding the Rate of Growth

Recognize that the growth rate of the percentage of babies born out of wedlock is \(0.6\% \) per year, starting from \(28\% \) in 1990.

Step 2: Formulating the Equation

Formulate a linear equation to model the situation. Let \(x\) denote the number of years since 1990 and \(y\) be the percentage of babies born out of wedlock. Then the equation can be written as \(y = 28 + 0.6x\) as the percentage was \(28\% \) in 1990 and it increases \(0.6\% \) every year.

Step 3: Solving the Equation

Solve the equation \(y = 40\) as we are interested in the year when \(40\% \) of babies will be born out of wedlock. This implies solving the equation \(40 = 28 + 0.6x\) for \(x\).

Step 4: Calculating the Year

After calculating \(x\), add it to 1990 to get the actual year.

Key Concepts

Percentage Increase

Percentage increase is a crucial concept when analyzing changes over time. It refers to the rate at which a certain quantity grows compared to its initial value. In real-life applications, it helps us understand trends like rising costs, population growth, or, as in our exercise, the increase in the percentage of babies born to unmarried parents.
To calculate a percentage increase, you first determine the initial and final values. The formula is:

• Determine the amount of change by subtracting the initial value from the final value.
• Divide this change by the initial value.
• Multiply the result by 100 to get a percentage.

For our exercise, the initial percentage is 28%, and it grows by 0.6% each year. This consistent growth allows us to predict future values using a linear equation. This process helps in decision-making and future planning by providing insights into the rate at which certain factors change.

Yearly Growth Rate

Yearly growth rate is the measure of how much a specific quantity increases year over year. It's often expressed as a percentage and is pivotal for comparing different periods or predicting future developments over time.
The exercise indicates a yearly growth rate of 0.6% for the percentage of babies born to unmarried parents. This number tells us the constant rate at which the percentage is increasing each year. Understandably, using this rate, we can create predictions about how this percentage will evolve over the years.
The growth rate acts like the 'slope' in a linear equation, where each year, the percentage climbs steadily. The concept of a constant yearly growth rate assumes the trend remains unchanged over the analyzed timeline. It's a straightforward yet powerful tool in forecasting future conditions based on past data.

Forecasting

Forecasting involves predicting future occurrences based on current and historical data. It's a vital process in various fields such as economics, environmental science, and social studies. In the exercise, forecasting predicts when 40% of babies will be born to unmarried parents.
To forecast accurately, you need to establish a model that depicts the current situation and anticipates future changes. In our case, the model is a simple linear equation that considers the starting percentage in 1990 and the annual increase of 0.6%.

• Start by identifying known data points, such as the percentage in a base year.
• Use the growth rate to project increases over time.
• Formulate an equation or use statistics to make predictions.

Forecasting with a linear model provides clear insights into when specific thresholds, like the 40% in the exercise, will likely be reached. This methodology is crucial for setting policies and making informed decisions about future trends.