Problem 117 - College Algebra

Question

The algebraic expression $$ \frac{73 t^{1 / 3}-28 t^{2 / 3}}{t} $$ describes the percentage of people in the United States applying for jobs \(t\) years after 1985 who tested positive for illegal drugs. Use a calculator to find the percentage who tested positive from 1986 through \(2001 .\) Round answers to the nearest hundredth of a percent. What trend do you observe for the percentage of potential employees testing positive for illegal drugs over time?

Step-by-Step Solution

Answer

After following steps 1 and 2, you will have the percentage of potential employees in the U.S. testing positive for illegal drugs each year from 1986 through 2001. Analyzing these percentages in step 3 helps you understand the trend over these years.

Step 1: Understand the Problem

The given algebraic expression is used to calculate the percentage of potential employees testing positive for illegal drugs. In this expression, \(t\) represents the number of years after 1985. Thus, we'll start from \(t=1\) (representing 1986) and go up till \(t=16\) (representing 2001).

Step 2: Calculate Percentages

Substitute each value of \(t\) from 1 to 16 into the given expression \(\frac{73 t^{1 / 3}-28 t^{2 / 3}}{t}\) and calculate the value. This will give you the percentage of potential employees testing positive for illegal drugs for each respective year.

Step 3: Observe Trend

Analyze the calculated percentages to find a trend over the time period 1986 to 2001. Consider whether the percentage is increasing, decreasing, or staying about the same over time. The trend could provide insights into the changes in drug use among potential employees in this time period.

Key Concepts

Algebraic Expression Evaluation

Evaluating algebraic expressions is a fundamental skill in college algebra, essential for solving various real-world problems. Algebraic expressions represent quantities in terms of variables and constants, combining them using operations such as addition, subtraction, multiplication, division, and exponentiation.

An expression like \(\frac{73 t^{1 / 3}-28 t^{2 / 3}}{t}\) involves exponents and rational exponents, which can be challenging for some students. Here's how to approach this evaluation:

• Understand the components: Identify the terms and their parts (e.g., numerical coefficients, variables, exponents).
• Substitute the variable: Replace \(t\) with the specific value representing years after 1985.
• Simplify the expression: Apply exponent rules, conduct the operations within the numerator, and divide by the denominator.
• Repeat for each year: Calculate the percentage for each year from 1986 to 2001.

The expression shows that the result depends on the number \(t\) raised to different fractional powers, which demonstrates how drug test positivity changes over time with respect to \(t\). It's crucial to review evaluating exponents and radicals to perform these calculations correctly.

Percentage Calculation

Percentage calculation is an invaluable tool for analyzing data and understanding proportions in various fields, including economics, statistics, and even healthcare. The percentage represents a part per hundred of a whole. In our exercise, we are focusing on the percentage of individuals who tested positive for illegal drugs.

To calculate these percentages, follow these steps:

• Compute the expression's value for each year.
• Ensure the result represents a portion of 100, as percentages are fractions of 100. If necessary, multiply by 100 - though in our case, the expression already gives us a percentage.
• Round the result to the nearest hundredth to comply with the problem's instructions, as precise values are vital in reporting statistical data.

Accurate percentage calculations are critical for data analysis and reporting, as they allow comparison across different scales and sizes, providing a clear picture of trends and changes in datasets.

Temporal Trend Analysis

Temporal trend analysis is examining data over time to identify patterns and changes. Such an analysis is essential for understanding dynamics in any field where time influences the data, such as the labor market, public health, or economics.

In this case, once we have computed the percentages for each year from 1986 through 2001, we look for the trend in these values:

• Arrange the data chronologically to make the trend more visible.
• Analyze how the percentage changes: Is it increasing, decreasing, or remaining steady?
• Use graphical representations, like line graphs, to visualize the trend if needed. This can make it easier to spot any patterns.
• Discuss potential causes for the trend and implications for the future.

Through temporal trend analysis, we can gain insights into how drug use among potential employees has evolved during a specific timeframe, which can help inform policy decisions and prevention strategies.