Question

The total amount I (in millions of dollars) of school expenditures from prekindergarten to a college level and the enrollment P (in millions) in prekindergarten through college in the United States can be modeled by $$ I=\frac{17,913 t+709,569}{1-0.028 t} \text { and } P=0.5906 t+70.219 $$ where \(t\) is the number of years since 2001. Find a model \(M\) for the annual education expenditures per student. Estimate the annual education expenditures per student in 2009. cant copy image

Short answer

Firstly, model M is formulated as \(M = \frac{\frac{17,913 t+709,569}{1-0.028 t}}{0.5906 t+70.219}\). Then, 't' value for the year 2009 is calculated as \(t = 2009 - 2001\). Substituting this into the model gives the annual education expenditure per student in 2009.

Step-by-step solution

Formulate the Model M

Start by noting that expenditure per student (M) is total expenditure (I) divided by total number of students (P). In algebraic form, it can be written as \(M = \frac{I}{P}\). Substituting the given equations \(I=\frac{17,913 t+709,569}{1-0.028 t}\) and \(P=0.5906 t+70.219\) leads to the model \(M = \frac{\frac{17,913 t+709,569}{1-0.028 t}}{0.5906 t+70.219}\). Simplify this to get the exact form of model M.

Estimate the Expenditure in 2009

To estimate the annual education expenditure per student in 2009, set \(t = 2009 - 2001\) in the model M formulated in step 1.

Evaluate and Simplify

Substitute the calculated value of 't' into model M and simplify it to find the annual education expenditure per student in 2009.

Key concepts

School Expenditures

Understanding school expenditures is critical for grasping the context of educational economics. These expenditures refer to the total amount of money spent by governments and institutions on education. This includes salaries for teachers and staff, infrastructure costs like building maintenance and construction, educational materials such as books, and technology required for teaching and learning. It's essential to differentiate between capital expenses, which are investments into long-term assets, and operational expenses, which cover the daily running costs.

Knowing how funds are allocated within the education system helps us evaluate the efficiency and prioritize areas that may need more investment. For instance, a rise in expenditures without corresponding improvements in student outcomes may indicate issues with how the funds are being utilized.

Education Expenditure Per Student

Education expenditure per student is a pivotal metric that puts the focus on individual student resources. It's obtained by dividing the total educational spendings by the total enrollment. This metric allows for a more nuanced analysis of the educational investment and can reveal disparities between different regions or educational institutions. It is also used as an indicator of quality or investment level in education. The higher the expenditure per student, the assumption is that the more resources are available per learner, potentially leading to better educational outcomes.

However, a higher spend per student doesn't necessarily translate directly into better educational quality, as it greatly depends on how effectively the resources are managed. This is why education economists also analyze the relationship between spending and academic achievement to assess the return on educational investments.

Algebraic Modeling

Algebraic modeling is a mathematical approach used to represent real-world scenarios with algebraic expressions or equations. In the context of education economics, it allows us to create models that can predict or analyze spending trends. Modeling helps in making informed decisions by anticipating future needs or trends based on past and present data.

In the exercise provided, the algebraic model connects the total school expenditures and enrollment to the time variable, 't', representing years since 2001. Through the model, we can estimate individual parameters like the annual education expenditure per student. This algebraic representation is powerful because, with it, we can forecast future expenditures or analyze the spending pattern over time, assisting policymakers and educational planners in their decision-making processes.