Question

The function \(I(n)=28,000+4,000 n\) represents the average annual income in dollars for a person with \(n\) years of college education. What is the best interpretation for the equation \(I(4)=44,000\) ? A. A person with 4 years of college should request an annual average salary of \(\$ 44,000\) when interviewing. B. A person with 4 years of college will earn \(\$ 44,000\) more each year on average than if they didn't attend college. C. A person with 4 years of college will earn \(\$ 44,000\) annually, on average. D. A person with 4 years of college should look for a position that starts at \(\$ 44,000\) annually.

Short answer

C: A person with 4 years of college will earn \( \$44000 \) annually, on average.

Step-by-step solution

Understand the function

The function given is \( I(n) = 28000 + 4000n \), where \( I(n) \) represents the average annual income in dollars for a person with \( n \) years of college education.

Substitute the value of \( n \)

To solve \( I(4) \), substitute \( n = 4 \) into the equation: \( I(4) = 28000 + 4000 \times 4 \).

Calculate the income

Compute the expression: \( I(4) = 28000 + 16000 = 44000 \). Thus, \( I(4) = 44000 \), meaning a person with 4 years of college education has an average annual income of \( \$44000 \).

Interpret the equation

The equation \( I(4) = 44000 \) indicates that the average annual income for a person with 4 years of college is \( \$44000 \).

Choose the best interpretation

Based on the calculated annual income, the best interpretation among the given options is C: A person with 4 years of college will earn \( \$44000 \) annually, on average.

Key concepts

Understanding Mathematical Functions

Mathematical functions are essential tools in various fields, including economics and education. A function is a relationship between inputs and outputs. In our exercise, we have the function \(I(n) = 28000 + 4000n\). Here, \(I(n)\) represents the average annual income (in dollars) based on \(n\) years of college education. Functions help us model real-world situations and allow us to predict outcomes based on given inputs. In our case, we can determine the average income for different years of college education.

Income Projections Using Functions

Income projections are necessary for understanding the potential financial benefits of different educational or professional decisions. Using the function \(I(n) = 28000 + 4000n\), we can project the average annual income for various levels of college education. For instance, substituting \(n = 4\) means we calculate \(I(4) = 28000 + 16000 = 44000\). This tells us that with four years of college education, the average annual income is \(\$44,000\). Such projections help individuals make informed decisions about investing time and resources in education, guiding them towards maximizing their potential earnings.

Valuing College Education

College education significantly impacts one's earning potential. The given function \(I(n) = 28000 + 4000n\) shows a direct relationship between the number of years of education and income. For every additional year of college, the average annual income increases by \(\$4,000\). This illustrates the financial advantage of extended education. Besides the increased earning potential, education often leads to personal and professional growth, better job opportunities, and improved quality of life. Understanding how education translates to tangible benefits, such as higher income, can motivate individuals to pursue and value higher education. By interpreting such functions, students can appreciate the long-term advantages of their educational choices.