Question

Federal Student Aid The total amount (in millions of dollars) of federal student aid disbursed in the years 1998 through 2005 can be approximated by \(453.11 x^{2}-5546.7 x+55,833\) where \(x\) represents the year, with \(x=8\) corresponding to 1998\. Evaluate the polynomial when \(x=14\) and \(x=15\). Then describe your results in everyday terms. (Source: U.S. Department of Education)

Short answer

After performing the computations, the total federal student aid corresponding to 2004 will be the value obtained from Step 2 and that for 2005 will be the value from Step 4. These values represent the total aid given in millions of dollars.

Step-by-step solution

Substitution of 'x' with 14

Start by substituting 'x' in the polynomial \(453.11 x^{2}-5546.7 x+55,833\) with 14 to represent the year 2004. This leads to the equation: \[453.11(14)^{2}-5546.7(14)+55,833\]

Performing the Calculations for 'x' equal to 14

Now, perform the calculations in the equation \[453.11(14)^{2}-5546.7(14)+55,833\]. Following the order of operations - parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right) - first square 14, then complete multiplication, before proceeding to subtraction, and finally the addition.

Substitution of 'x' with 15

Next, substitute 'x' in the polynomial \(453.11 x^{2}-5546.7 x+55,833\) with 15 to represent the year 2005. This leads to the equation: \[453.11(15)^{2}-5546.7(15)+55,833\]

Performing the Calculations for 'x' equal to 15

Perform the calculations in the equation \[453.11(15)^{2}-5546.7(15)+55,833\]. As before, follow the order of operations. First square 15, then complete the multiplication. Finally proceed to subtraction, and addition.

Interpreting the Results

The values obtained from the calculations represent the total amount (in millions of dollars) of Federal Student Aid disbursed in the years 2004 and 2005 respectively. The increase or decrease in the amount disbursed can be outlined by comparing the two values - a higher value for 2005 indicates an increase in aid, whereas a lower value suggests a decrease.

Key concepts

Federal Student Aid

The Federal Student Aid program is designed to assist students in covering college-related expenses, easing the financial burden of higher education. The U.S. Department of Education oversees this program, ensuring eligible students receive the aid they need. This program is crucial as it enables millions of students to attend college who might not otherwise afford it. By evaluating the disbursed aid over time, we gain insights into the program's reach and effectiveness. For instance, the exercise focuses on calculating the aid disbursed in specific years using a polynomial expression. Through these calculations, we can interpret trends in the funding, which in turn reflects the program's impact and adjustments based on student needs or policy changes.

Order of Operations

When solving any mathematical problem, it's essential to follow the order of operations. This sequence is commonly remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Let’s break it down:

• First, do any calculations inside parentheses.
• Next, evaluate exponents or powers.
• After that, perform all multiplication and division steps from left to right.
• Lastly, handle addition and subtraction from left to right.

This systematic approach ensures that everyone solves the problem consistently and accurately. In the exercise, following the order of operations allows us to correctly evaluate how much federal student aid was disbursed, ensuring accuracy in our final result.

Quadratic Function

A quadratic function is a polynomial equation of the form \( ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. These functions are crucial in mathematical modeling because they can describe a wide range of real-world phenomena. The function in the exercise, \( 453.11x^2 - 5546.7x + 55,833 \), is specifically a quadratic function.
In this context, the quadratic function is being used to approximate the federal student aid disbursements over a series of years. This type of function can model changes over time, capturing the increase or decrease in aid with each passing year. Quadratic functions have a distinctive parabolic graph, making them useful for understanding trends and changes that are not constant over time.

Mathematical Substitution

Substitution in mathematics involves replacing a variable with a given value. It's a key technique used to solve equations or evaluate expressions. In our exercise, substitution was used to replace \( x \) with specific values representing years. This allows us to put a specific time frame into the formula to find the outcome associated with that year.
For example, when \( x = 14 \) is substituted into the quadratic expression, we attribute this to the year 2004, facilitating the calculation of federal aid for that year. Substitution makes it possible to compute outcomes directly, saving time and reducing the potential for error. By substituting different values, we can effectively evaluate and compare results across multiple scenarios or time periods.