Problem 220 - Precalculus

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 rectangle as a function of \(x .\)

Step-by-Step Solution

Answer

The polynomial function is \( A(x) = 2x^2 + 26x + 60 \).

Step 1: Identify the initial dimensions of the rectangle

The problem states that the rectangle's initial length is 10 inches and its initial width is 6 inches.

Step 2: Determine the new dimensions

The length of the rectangle is increased by \(x\) inches, resulting in a new length of \(10 + x\) inches. The width is increased by twice that amount, or \(2x\) inches, resulting in a new width of \(6 + 2x\) inches.

Step 3: Write the expression for the updated area

The area of a rectangle is given by the product of its length and width. Therefore, the area \(A(x)\) as a function of \(x\) can be expressed as:\[ A(x) = (10 + x)(6 + 2x) \]

Step 4: Expand the expression for area

Use the distributive property to expand the expression for the area:\[A(x) = (10 + x)(6 + 2x) = 10(6 + 2x) + x(6 + 2x)\]

Step 5: Simplify the expanded expression

Continue expanding and combine like terms:\[A(x) = 10 \cdot 6 + 10 \cdot 2x + x \cdot 6 + x \cdot 2x = 60 + 20x + 6x + 2x^2\]Combine the terms:\[A(x) = 2x^2 + 26x + 60\]

Step 6: Present the polynomial function

Therefore, the polynomial function representing the area of the rectangle as a function of \(x\) is:\[ A(x) = 2x^2 + 26x + 60 \]

Key Concepts

Rectangle Area

The concept of area is fundamental when dealing with geometric shapes, such as rectangles. The area of a rectangle is the measure of the space it occupies in a two-dimensional plane. It is calculated by multiplying the length by the width. For example, the area of a rectangle with a length of 10 inches and a width of 6 inches is simply:

• Area = Length × Width = 10 inches × 6 inches = 60 square inches

When dimensions change, the area changes too. In the exercise, when the length and width increase by certain amounts, a new area must be calculated accordingly. The task is to express this new area as a function of a variable, demonstrating how algebra can be used in practical geometry problems.

Function Representation

A function, in mathematics, represents a relationship between inputs and outputs. It allows us to understand how changes in one quantity affect another.
In the rectangle problem, we are asked to represent the area as a function of an increase in length, denoted by the variable \(x\). The exercise clearly states that the length is increased by \(x\) inches while the width increases by twice that amount, or \(2x\) inches.
Thus, the new function representing the area, \(A(x)\), incorporates these changes in dimensions:

• New Length = 10 + x
• New Width = 6 + 2x
• Area Function = \(A(x) = (10 + x)(6 + 2x)\)

This function shows how the area varies with changes in \(x\), modeling the dynamic nature of dimensions.

Distributive Property

The distributive property is an essential algebraic property that allows us to multiply a single term by terms inside a parenthesis. It simplifies calculations and helps in expanding expressions.
For example, in the given problem, we need to expand the expression for the area, \((10 + x)(6 + 2x)\).
Applying the distributive property, we proceed as follows:

• Distribute \(10\) over \((6 + 2x)\): \(10 \, \cdot \, 6 + 10 \, \cdot \, 2x\)
• Distribute \(x\) over \((6 + 2x)\): \(x \, \cdot \, 6 + x \, \cdot \, 2x\)
• Combine to expand: \(60 + 20x + 6x + 2x^2\)

The resulting expression can then be simplified by combining like terms, ultimately indicating the expanded form as a polynomial.

Algebraic Expressions

An algebraic expression consists of variables, constants, and operations (like addition and multiplication) combined to represent a value or a relationship.
Expressions can be simplified and manipulated to suit various requirements, such as solving problems or modeling scenarios.
In this rectangle exercise, the steps led us to form a polynomial expression for the area:

• Combine terms: \(60 + 20x + 6x + 2x^2\)
• Simplify it to: \(60 + 26x + 2x^2\)

This results in the polynomial function, \(A(x) = 2x^2 + 26x + 60\), which concisely represents how the area changes with \(x\).
Learning to work with such expressions builds foundational skills in algebra and helps students model real-world problems.