Question

The average yearly cost $C$ of higher education at public institutions in the United States for the academic years $1995/1996$ to $2004/2005$ can be modeled by $C=30.57t^2-259.6t+6828,6\le t\le 15$ where $t$ represents the year, with $t=6$ corresponding to the $1995/1996$ school year (see figure). Use the model to predict the academic year in which the average yearly cost of higher education at public institutions exceeds $\mathrm{\$}12,000$.

Short answer

Following the above steps, the calculation will give you two solutions for $t$. One of them will be outside the range of [6, 15] so you can discard it. The valid solution corresponds to the academic year when the average yearly cost of higher education at public institution exceeds $12,000.

Step-by-step solution

Set up the inequality

The problem is asking for when the cost $C$ exceeds $12,000. Therefore, set up the equation with the given cost function: $\mathrm{\$}12,000=30.57t^2-259.6t+6828$.

Rearrange the equation

Arrange the equation so that it equals to zero: $30.57t^2-259.6t+6828-\mathrm{\$}12,000=0$. Upon simplification, this gives: $30.57t^2-259.6t-5112=0$. Note that we subtract $\mathrm{\$}12,000$ from $6828$ yielding $-5112$.

Solve for $t$

This is a quadratic equation in the form of $a{x}^2+bx+c=0$, and it can be solved by using the quadratic formula $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Plugging in the values gives two solutions for $t$. Remember that $t$ is representing years beyond 1990, and therefore $t$ must be a positive value.

Assess the validity of solutions

According to the model, $t$ is within the interval [6, 15]. Therefore, choose $t$ which is within this interval and discard the one which is not.

Key concepts

Quadratic Equations

Understanding quadratic equations is essential not only in algebra but also in various applications, such as predicting costs and analyzing trends. A quadratic equation is a second-degree polynomial typically written in the form of $a{x}^2+bx+c=0$ where $a$, $b$, and $c$ are constants, and $x$ represents an unknown variable.

When faced with a quadratic equation while predicting higher education costs, as in the provided exercise, we need to solve for $t$ to find the year when costs exceed a certain amount. To begin, you'll first set the calculated cost equal to your target amount, rearrange the equation, and simplify it so that it's set to zero, adhering to the form $a{x}^2+bx+c=0$.

Solving for $x$—or $t$ in our context—generally involves the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. The solution to the equation provides the potential values for $t$, which then should be analyzed in the context of the exercise to determine the realistic year when costs exceed the specified amount. It's crucial to remember that the discriminant ($b^2-4ac$ part under the square root) determines the nature of the roots, implying how many real solutions we can expect.

Inequalities in Algebra

Inequalities are algebraic expressions that tell us about the relative size of two values. In algebra, inequalities are used to compare numbers or expressions, using signs like $>$ (greater than), $<$ (less than), $\ge$ (greater than or equal to), and $\le$ (less than or equal to).

In the context of predicting higher education costs, we set up an inequality to determine when the cost exceeds $12,000. In this case, we are seeking a solution for $t$ such that $30.57t^2-259.6t+6828>12,000$. Once we rearrange the inequality to create a standard quadratic equation, like in Step 2 of the provided solution, we solve it as we would any other quadratic equation.

Solving inequalities sometimes involves identifying ranges of values rather than exact numbers. This is evident when we consider the interval $[6,15]$ for $t$, based on the data from the modeling exercise. The solutions to the inequality will show us the period when education costs are projected to surpass the threshold, adhering to the constraints of the model.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world scenarios in order to analyze and predict behaviors or outcomes. These models can take various forms, such as equations or simulations, and in education economics, they help in predicting costs over time.

The equation $C=30.57t^2-259.6t+6828$ provided in the exercise represents a mathematical model of the growth of education costs, where $t$ is the number of years since the 1995/1996 academic year. This kind of model helps in visualizing and forecasting future changes in cost based on historical data.

Key to effective mathematical modeling is the ability to validate and interpret the results within the context of the real world. The model may have limitations, such as a particular time frame ($6\le t\le 15$ in our case). It makes assumptions based on historic trends, and while it may provide valuable insights, it is always an approximation. The predictive power of the model should always be carefully assessed, keeping in mind that it is a tool to aid understanding and decision making, not an absolute predictor of future events.