Problem 80 - Contemporary Precalculus

Question

The percentage of people 25 years old and older who have a Bachelor's degree or higher was about 25.6 in 2000 and 27.7 in 2004 (a) Find a linear equation that gives the percentage of people 25 and over who have a Bachelor's degree or higher in terms of time \(t\), where \(t\) is the number of years since 2000 . Assume that this equations remains valid in the future. (b) What will the percentage be in \(2010 ?\) (c) When will \(34 \%\) of those 25 and over have a Bachelor's degree or higher?

Step-by-Step Solution

Answer

Short answer: The linear equation that relates the percentage of people 25 and older with a Bachelor's degree or higher to the years is \(y = 0.525x + 25.6\), where \(x\) is the number of years since 2000. In 2010, the percentage is approximately \(30.85\%\), and the percentage will reach \(34\%\) in the year 2016.

Step 1: Find the slope of the line

The first step is to find the slope of the line connecting the two given points: (0, 25.6) representing year 2000 and percentage 25.6%, and (4, 27.7) representing year 2004 and percentage 27.7%. The slope formula is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where \(m\) is the slope, \((x_1, y_1)\) are the coordinates of the first point, and \((x_2, y_2)\) are the coordinates of the second point. Now substitute the given points: \(m = \frac{27.7 - 25.6}{4 - 0}\)

Step 2: Calculate the slope

By calculating the slope, we get: \(m = \frac{2.1}{4} = 0.525\) The slope of the line is \(0.525\). This means that, on average, the percentage of people 25 and older with Bachelor's degree or higher will increase by \(0.525 \%\) per year.

Step 3: Find the intercept of the line

Now we need to find the intercept of the line. We can use the point-slope form of the linear equation: \(y - y_1 = m(x - x_1)\) We already have the slope, \(m=0.525\), and we can use the point from year 2000, \((0, 25.6)\), to find the intercept: \(y - 25.6 = 0.525(x - 0)\) After simplifying, we have: \(y = 0.525x + 25.6\)

Step 4: Answer question (a)

We found the linear equation that gives the percentage of people 25 and over who have a Bachelor's degree or the higher in terms of time \(t\): \(y = 0.525x + 25.6\) where \(x\) is the number of years since 2000.

Step 5: Answer question (b)

To find the percentage in 2010, we need to substitute \(x=10\) (since 2010 is 10 years after 2000) into our equation: \(y = 0.525(10) + 25.6\) \(y = 5.25 + 25.6\) \(y = 30.85\) In 2010, the percentage of people 25 and over with a Bachelor's degree or higher will be approximately \(30.85\%\).

Step 6: Answer question (c)

To find when the percentage will be \(34\%\), we need to set the equation equal to 34 and solve for \(x\): \(34 = 0.525x + 25.6\) Now, we subtract 25.6 from both sides: \(8.4 = 0.525x\) Lastly, we need to solve for \(x\): \(x = \frac{8.4}{0.525} = 16\) \(x = 16\) The percentage will reach \(34\%\) after 16 years (since 2000), which will be in the year 2016.

Key Concepts

Slope Calculation

The slope of a line in a graph represents how steep the line is and tells us the rate of change between two points. To calculate the slope (\(m\)) in linear equations, we use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points on the line.
In our example, for the years 2000 and 2004, we used \((0, 25.6)\) and \((4, 27.7)\) as points. This gave us:

• The difference in percentage: \(27.7 - 25.6 = 2.1\)
• The difference in years: \(4 - 0 = 4\)

So, the slope is:\[m = \frac{2.1}{4} = 0.525\]This means every year, the percentage of people with Bachelor's degrees increases by \(0.525\%\).Understanding slope helps us see the pace of change in data trends. It is fundamental to linear equations as it directly influences projections and predictions.

Intercept Determination

The intercept in a linear equation is the point where the line crosses the y-axis. It's a key value because it shows the starting point—or initial value—when the variable \(x\) (time in our case) is zero.
The formula for the line using slope and intercept is:\[y = mx + b\]Here, \(b\) represents the intercept. Using the point-slope form \((y - y_1 = m(x - x_1))\) and substituting the point \((0, 25.6)\), the equation is:

• \[y - 25.6 = 0.525(x - 0)\]
• Simplifying gives: \[y = 0.525x + 25.6\]

Thus, the intercept \(b\) is \(25.6\), indicating the percentage in the year 2000 was \(25.6\%\).
Knowing the intercept helps us define the linear equation fully, making it easier to project data trends over time.

Mathematical Modelling

Mathematical modelling involves using equations to represent real-world scenarios. Linear equations like \[y = mx + b\] help us model trends based on given data points.
In the exercise, we created a model showing how the percentage of people with Bachelor's degrees changes over time.
Why is this important?

• Models simplify complex systems, showing relationships clearly.
• They aid in making predictions based on trends.
• Models allow testing of different scenarios, showing possible outcomes.

With the equation \[y = 0.525x + 25.6\], we connected education attainment to time, offering a clear picture of the increase in educational achievements. Mathematical modelling serves as a crucial tool in understanding and predicting real-life situations.

Projection and Prediction

Projection involves extending current trends into the future to predict outcomes. Using our linear equation model, \(y = 0.525x + 25.6\), we can forecast future educational attainment rates.
Let's apply it:

• To find the percentage in 2010, substitute \(x = 10\):\[y = 0.525 \times 10 + 25.6 = 30.85\%\]
• To determine when \(34\%\) will be reached:Set \(y = 34\) and solve:\[34 = 0.525x + 25.6\]\[x = \frac{8.4}{0.525} = 16\]
• This results in the year 2016.

Projections help in planning and policymaking as we anticipate future needs or outcomes. By reliably extending trends, we can better prepare for and adapt to changes in various sectors.