Problem 68 - College Algebra

Question

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

Step 1: Understand Initial Dimensions

Initially, the rectangle has a length of 10 inches and a width of 6 inches. This is our starting point.

Step 2: Apply Changes to Dimensions

The length is increased by \(x\) inches, making the new length \(10 + x\). The width is increased by twice that amount, making the new width \(6 + 2x\).

Step 3: Establish the Area Function

The area \(A\) of a rectangle is calculated as the product of its length and width. Thus, the area \(A(x)\) is \((10 + x)(6 + 2x)\).

Step 4: Expand the Polynomial

Expand the expression \((10 + x)(6 + 2x)\) using the distributive property: \[A(x) = 10(6 + 2x) + x(6 + 2x) = 60 + 20x + 6x + 2x^2.\]

Step 5: Simplify the Expression

Combine like terms in the polynomial: \[A(x) = 2x^2 + 26x + 60.\]

Step 6: Conclude with the Polynomial Function

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

Key Concepts

Area of a Rectangle

The concept of the area of a rectangle is fundamental in geometry. To find the area of a rectangle, we simply multiply the length by the width. This gives us a measure of the two-dimensional space within the rectangle's boundaries.
For instance, if a rectangle has a length of 10 inches and a width of 6 inches, the area would be calculated as:

• Area = 10 inches * 6 inches

This results in an area of 60 square inches. When either the length, the width, or both are changed, the area will change accordingly. It's a direct relationship. Whenever you increase the dimensions of a rectangle, the area's numerical value also increases, assuming all other dimensions remain positive.

Distributive Property

The distributive property is a useful principle in algebra that helps simplify complex expressions. It states that multiplying a sum by a number is the same as multiplying each addend individually by the number, and then adding the products. In formula terms, this is written as:

• \( a(b + c) = ab + ac \)

In the context of our rectangle problem, we need to find the area of a constructed rectangle using the new dimensions, \( (10 + x) imes (6 + 2x) \).

Expanding Expressions

Expanding expressions involves applying the distributive property to remove parentheses and simplify an equation. Take our polynomial from the rectangle problem:

• \( (10 + x)(6 + 2x) \)

Using the distributive property, we calculate:

• \( 10(6 + 2x) + x(6 + 2x) \)

This yields four individual multiplications:

• \( 10 * 6 \) = 60
• \( 10 * 2x \) = 20x
• \( x * 6 \) = 6x
• \( x * 2x \) = 2x^2

Gathering these together, the expanded polynomial takes the form:

• \( 2x^2 + 20x + 6x + 60 \)

After combining like terms, we simplify this to:

• \( 2x^2 + 26x + 60 \)

This is the resulting polynomial function that represents the area of the rectangle as a function of \( x \). Each coefficient and term tells a part of the story of how changes in dimensions affect the area.