Problem 22 - College Algebra

Question

The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given in Table \(4^{|14|}\) Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the percentage exceed 35\(\% ?\) $$\begin{array}{ccccccccc}\hline{\text { Year }} & {1990} & {1992} & {1994} & {1996} & {1998} & {2000} & {2002} & {2004} & {2006} & {2008} \\ \hline {\text { Percent Graduates }} & {21.3} & {21.4} & {22.2} & {24.6} & {24.4} & {25.6} & {26.7} & {27.7} & {28} & {29.4} \\ \hline\end{array}$$

Step-by-Step Solution

Answer

If the trend continues, the percentage will exceed 35% around 2020.

Step 1: Calculate Differences

First, we calculate the differences in the percentage of college graduates between consecutive years. - From 1990 to 1992: 21.4 - 21.3 = 0.1 - From 1992 to 1994: 22.2 - 21.4 = 0.8 - From 1994 to 1996: 24.6 - 22.2 = 2.4 - From 1996 to 1998: 24.4 - 24.6 = -0.2 - From 1998 to 2000: 25.6 - 24.4 = 1.2 - From 2000 to 2002: 26.7 - 25.6 = 1.1 - From 2002 to 2004: 27.7 - 26.7 = 1.0 - From 2004 to 2006: 28 - 27.7 = 0.3 - From 2006 to 2008: 29.4 - 28 = 1.4 The differences are inconsistent, suggesting that the trend may not be perfectly linear.

Step 2: Identify Linear Segments

Examine if there are segments where changes appear consistent. Notice that from 2000 to 2008, the changes are more consistent (1.1, 1.0, 0.3, 1.4). Although not perfectly linear, these changes suggest a relatively steady increase around approximately 1% per 2 years.

Step 3: Estimate the Linear Trend

Assuming a consistent average yearly increase rate from 2000 onwards, we look for an average increase:Average from 2000, 2002, 2004, 2006, 2008: \[ \text{Average Increase} = \frac{1.1 + 1.0 + 0.3 + 1.4}{4} \approx 0.95 \]The rate per year is approximately half of that (since these changes are bi-annual):\[ \text{Rate per Year} = \frac{0.95}{2} \approx 0.475 \]

Step 4: Project Future Values

Starting in 2008, with 29.4% graduates, project forward assuming the trend continues:Let's represent future years as variable \(x\), where \(x\) is the number of years beyond 2008. We want \(29.4 + 0.475x > 35\). Solve for \(x\):\[ 29.4 + 0.475x > 35 \]\[ 0.475x > 5.6 \]\[ x > \frac{5.6}{0.475} \approx 11.79 \]Thus, it will take approximately 12 years to exceed 35%.

Step 5: Calculate the Exact Year

Since \(x\) represents the number of years beyond 2008, we add 12 years:\[ 2008 + 12 = 2020 \]So, under the assumption of a linear trend, the percentage of college graduates will likely exceed 35% in the year 2020.

Key Concepts

Data Analysis

In the realm of data analysis, one of the key tasks is to examine data over time to recognize patterns and trends. The U.S. Census data provided illustrates the percentage of the population aged 25 years or older who are college graduates, tracked over a series of years. By analyzing this data, we look to identify whether these percentages follow a recognizable pattern. The process often begins with examining raw data for consistency or notable deviations.

Differences in values from year to year are calculated to assess variability. A consistent pattern in these differences could suggest a linear relationship. In this case, the differences between consecutive years appear somewhat erratic but discerning a trend requires a closer look.

• Examining increments between data points helps analysts determine if a stable rate of change is apparent.
• Assessing longer intervals or segments within the dataset can reveal underlying linear trends, even if universal consistency is absent.

By scrutinizing the data from 2000 onward, a more noticeable trend emerges, indicating an overall increase despite minor fluctuations.

Rate of Change

Understanding the rate of change is crucial for determining the progress depicted by a dataset over specified time intervals. The rate of change indicates how quickly or slowly a certain variable is changing compared to another, often time. In the context of our problem, we calculate it to discern the annual growth in the percentage of college graduates.

The differences between each set of consecutive years offer insights. For data that appears somewhat inconsistent initially, focusing on specific segments can offer clearer rates:

• The differences between 2000 to 2008 range from 1.1% to 1.4% biannually.
• Calculating the average increase across these years, we find it to be approximately 0.95% biannually.

This implies the need to determine the annual rate, requiring the average to be halved.
Hence, an increase of approximately 0.475% per year supplies a reasonable approximation of growth under the assumption of linearity. By maintaining focus on these calculations, analysts can project future achievements based on historical data rates.

Projections

Once a trend and rate of change are established, we move towards creating projections. Projections extend the data analysis into the future to estimate upcoming trends and values. For this dataset, we used the calculated rate of change to foresee the percentage of college graduates exceeding 35%.

To make these projections:

• We begin with the most recent data point, in this case, 29.4% from 2008.
• The linear trend continues with a known rate of change, 0.475% annually.

If our goal is to determine when the graduate percentage exceeds 35%, we set up an inequality to find the required years:\[29.4 + 0.475x > 35\]This is solved to derive the variable \(x\), representing the years needed beyond 2008 to surpass the threshold.
With calculations showing that approximately 12 years are needed, we predict the percentage will exceed 35% by 2020.
Such projections are invaluable for planning in various domains, guiding stakeholders in education or government sectors as they prepare for anticipated changes.