Problem 82 - College Algebra and Calculus: An Applied Approach

Question

College Costs The average yearly cost \(C\) of attending a private college full time for the academic years \(1999 / 2000\) to \(2004 / 2005\) in the United States can be approximated by the model \(C=45.6 t^{2}+15,737, \quad 10 \leq t \leq 15\) where \(t=10\) corresponds to the \(1999 / 2000\) academic year (see figure). Use the model to predict the year in which the average cost of attending a private college full time is about \$30,000. (Source: U.S. National Center for Education Statistics)

Step-by-Step Solution

Answer

The year when the average cost of attending a private college full time is predicted to be about $30,000 is approximately the 2006/2007 academic year.

Step 1: Assign the given cost value to the model

In order to find the year when the cost will reach $30,000 we need to replace \(C\) in the equation with 30000, like this: \[30000 = 45.6t^2 + 15737\]

Step 2: Solve the equation for \(t\)

To isolate \(t\) on one side of the equation, subtract 15737 from both sides to obtain: \[14263 = 45.6t^2\]. Then divide both sides by 45.6 which results in the equation: \[t^2 = 14263 / 45.6\]. To solve for \(t\) take the square root of both sides.

Step 3: Find the corresponding year

After calculating \(t\), we add the result to 1999 since the problem specifies \(t = 10\) corresponds to the 1999/2000 academic year. This gives the academic year in which the cost of attending college is predicted to be $30,000.

Key Concepts

Quadratic Equations

When tackling problems involving college cost predictions, we often use quadratic equations. These are polynomial equations of the second degree, which means they have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In this context, the variable \(t\) in the model \(C=45.6t^2+15,737\) takes the place of \(x\), and we're looking at the relationship between time and cost. Solving quadratic equations often involves isolating the variable (which represents time in our problem) and finding its value by taking the square root of both sides of the equation after simplifying it. Knowing how to manipulate these equations is essential for predicting future college costs based on established trends.

College Tuition Trends

Observing college tuition trends over the years allows us to predict future costs, an important factor for long-term educational planning. The trends show that college tuition has been increasing consistently, helping us to model these changes mathematically. The equation provided in the exercise reflects this trend in a simplified form, using historical data to forecast future costs. A clear understanding of these trends and their patterns can be extremely valuable for students and their families who are planning for the financial commitment of higher education.

Mathematical Models in Economics

In economics, mathematical models are frequently employed to make sense of complex financial data. These models use equations to describe the relationship between different variables, such as time and cost in the college cost prediction model. The quadratic model we use here is a simple yet powerful tool that illustrates how costs might evolve over time. It factors in historical data and projects it into the future. Despite their utility, it's important to recognize the limitations of these models—they are based on assumptions and past trends, which do not always accurately predict future conditions.

Academic Year Cost Analysis

A critical part of weighing the financial burden of college education is conducting an academic year cost analysis. This involves breaking down not only tuition fees but also other costs such as room, board, books, and other expenses into an annual figure. Using the quadratic equation provided in the exercise allows for a theoretical snapshot of how tuition itself is projected to change over the years, which is an integral piece of the overall cost analysis puzzle. While the focus here is on the tuition aspect, a comprehensive view of all college-related expenses would provide a more realistic picture of the full financial commitment for each academic year.