Skip to main content
Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 25
Chapter 2, Problem 25

Dimensions of a Rug. Zachary wants to buy a rug for a room that is 12 ft wide and 15 ft long. He wants to leave a uniform strip of floor around the rug. He can afford to buy 108 ft2 of carpeting. What dimensions should the rug have?
Top-down view of a 15 by 12 ft room with a decorative rug and 108 square feet of carpeting area shown.

Verified step by step guidance
1
Let the width of the uniform strip around the rug be represented by \(x\) feet. Since the room is 12 ft wide and 15 ft long, the dimensions of the rug will be reduced by \$2x\( on each side, making the rug's width \)(12 - 2x)\( and length \)(15 - 2x)$.
Write an expression for the area of the rug in terms of \(x\): \(\text{Area} = (12 - 2x)(15 - 2x)\).
Set the area expression equal to the amount of carpeting Zachary can afford, which is 108 square feet: \((12 - 2x)(15 - 2x) = 108\).
Expand the left side of the equation to form a quadratic equation: \(180 - 24x - 30x + 4x^2 = 108\).
Simplify and rearrange the quadratic equation to standard form: \(4x^2 - 54x + 72 = 0\). Then, solve for \(x\) using the quadratic formula or factoring to find the width of the strip, and subsequently find the rug's dimensions.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
11m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Area of a Rectangle

The area of a rectangle is found by multiplying its length by its width. In this problem, both the room and the rug are rectangular, so understanding how to calculate area is essential to relate the rug's dimensions to the amount of carpeting Zachary can afford.
Recommended video:
Guided course
6:19
Systems of Inequalities

Uniform Strip Around the Rug

A uniform strip means the rug is placed so that the distance from the rug's edges to the room's walls is the same on all sides. This concept helps set up expressions for the rug's length and width in terms of the strip width, which is key to forming an equation.
Recommended video:
5:12
Graph Ellipses at Origin

Forming and Solving Quadratic Equations

Setting up the rug's area as a function of the strip width leads to a quadratic equation. Solving this equation allows us to find the strip width and thus the rug's dimensions. Understanding how to form and solve quadratics is crucial for finding the correct solution.
Recommended video:
06:08
Solving Quadratic Equations by Factoring
Related Practice
Textbook Question

Perform each operation. Write answers in standard form. (5-11i)(5+11i)

1
views
Textbook Question

Perform each operation. Write answers in standard form. (-8+2i)(-1+i)

Textbook Question

Solve each inequality. Give the solution set in interval notation. -(2/3)x-(1/6)x+(2/3)(x+1)≤4/3

Textbook Question

Solve each equation. -4(2x-6) +8x= 5x+24+x

Textbook Question

Solve each equation. 3/(x2+x-2) - 1/(x2-1) = 7/(2x2+6x+4)

4
views
Textbook Question

Solve each problem. See Example 2. At the 2008 Summer Olympics in Beijing, Usain Bolt set a new Olympic and world record in the 100-m dash with a time of 9.69 sec. If this pace could be maintained for an entire 26-mi marathon, what would his time be? How would this time compare to the fastest time for a marathon, which is 2 hr, 3 min, 23 sec, set in 2013? (Hint: 1 m ≈ 3.281 ft.) (Data from Sports Illustrated Almanac.)

1
views