Problem 68 - College Algebra with Corequisite Support

Question

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by \(x\) inches and the width increased by twice that amount, express the area of the rectangle as a function of \(x\).

Step-by-Step Solution

Answer

The area as a function of \(x\) is \(A(x) = 2x^2 + 26x + 60\).

Step 1: Identify Rectangle Dimensions

The original dimensions of the rectangle are given as a length of 10 inches and a width of 6 inches.

Step 2: Determine New Dimensions

When the length is increased by \(x\) inches, it becomes \(10 + x\) inches. Similarly, when the width is increased by twice that amount, or \(2x\), the new width becomes \(6 + 2x\) inches.

Step 3: Describe Area Formula

The area \(A\) of a rectangle is given by the formula \(A = \text{length} \times \text{width}\).

Step 4: Substitute New Dimensions into Area Formula

Substitute the new dimensions into the area formula: \(A(x) = (10 + x)(6 + 2x)\).

Step 5: Expand the Expression

Expand the product: \(A(x) = (10 + x)(6 + 2x) = 10 \cdot 6 + 10 \cdot 2x + x \cdot 6 + x \cdot 2x\).

Step 6: Simplify the Expanded Expression

Simplify the expression by combining like terms: \(A(x) = 60 + 20x + 6x + 2x^2 = 2x^2 + 26x + 60\).

Step 7: Final Polynomial Function

The polynomial function that represents the area of the rectangle as a function of \(x\) is \(A(x) = 2x^2 + 26x + 60\).

Key Concepts

Understanding Rectangle Dimensions

When dealing with problems involving rectangles, it's essential to clearly grasp the concept of rectangle dimensions. A rectangle, in its most basic sense, has two distinct dimensions: length and width. These dimensions are crucial for any calculations related to area or perimeter. In this exercise, we start with a rectangle with a length of 10 inches and a width of 6 inches. These measurements form the basis of our problem. To change the dimensions, an increase is introduced: the length is augmented by an unknown quantity, represented by the variable \( x \). The width, however, is augmented by twice this amount, letting us explore polynomial expressions and their applications.By comprehending how each dimension can change independently or dependently, you're laying the foundation for more complex calculations. This understanding is pivotal when manipulating formulas and developing solutions.

Calculating the Rectangle's Area

The area of a rectangle is a fundamental concept in geometry and is calculated using the area formula. The standard expression for the area \( A \) of a rectangle is given by:\[ A = \, \text{length} \, \times \, \text{width} \] In this exercise, we have modified dimensions to plug into this formula. The length becomes \(10 + x\), and the width changes to \(6 + 2x\). Plugging these into the formula, we establish that:\[ A(x) = (10 + x)(6 + 2x) \] This equation captures how the rectangle's area changes with respect to \(x\), demonstrating the interplay of polynomial functions. Applying the area formula is a practical demonstration of approaching more advanced algebraic and geometric problems, establishing a step toward broader algebraic proficiency.

Expanding and Simplifying Expressions

After establishing the polynomial expression for the rectangle's area, the next critical step involves expanding and simplifying this expression. Expanding involves distributing each term in the first binomial across each term in the second binomial:\[ A(x) = (10 + x)(6 + 2x) = 10 \cdot 6 + 10 \cdot 2x + x \cdot 6 + x \cdot 2x \] This results in four individual products: \(10 \cdot 6\), \(10 \cdot 2x\), \(x \cdot 6\), and \(x \cdot 2x\). Once calculated, these become:- 60- 20x- 6x- 2x²The final task is to simplify by combining like terms in the expression: the \(x\) terms are summed together, yielding:\[ A(x) = 2x^2 + 26x + 60 \] This expanded and simplified polynomial \(A(x)\) now beautifully represents the area as a function of \(x\). Expanding expressions efficiently is an invaluable skill for handling polynomial functions, enabling students to tackle even complex equations with confidence.