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Chapter 5: Problem 68

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 \(x .\)

Short Answer

Expert verified
The area function is \(A(x) = 2x^2 + 26x + 60\).

Step by step solution

01

Understand Initial Dimensions

Initially, the rectangle has a length of 10 inches and a width of 6 inches. This is our starting point.
02

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 \(6 + 2x\).
03

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 \((10 + x)(6 + 2x)\).
04

Expand the Polynomial

Expand the expression \((10 + x)(6 + 2x)\) using the distributive property: \[A(x) = 10(6 + 2x) + x(6 + 2x) = 60 + 20x + 6x + 2x^2.\]
05

Simplify the Expression

Combine like terms in the polynomial: \[A(x) = 2x^2 + 26x + 60.\]
06

Conclude with the Polynomial Function

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. \(f(x)=12 x^{4}+55 x^{3}+12 x^{2}-117 x+54\)

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and \(z\). When \(x=4\) and \(z=2\), then \(y=16\). Find \(y\) when \(x=3\) and \(z=3\).

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly as the square root of \(x\) and when \(x=36, y=2\).

Use Descartes鈥� Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=2 x^{4}-5 x^{3}-5 x^{2}+5 x+3\)

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with \(y\) -coordinates given. $$f(x)=x^{3}+8 x-4, y=-1,0,1$$
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