Solve each problem. Zachary wants to buy a rug for a room that is 12 ft wide and 15 ft long. He

step 1 Okay, we're given a room. That's 15 by 12 feet, and we're going to place a rug in the middle, an

Solve each problem. Zachary wants to buy a rug for a room...

Solve each problem.
Zachary wants to buy a rug for a room that is $12 \mathrm{ft}$ wide and $15 \mathrm{ft}$ long. He wants to leave a uniform strip of floor around the rug. He can afford to buy $108 \mathrm{ft}^{2}$ of carpeting. What dimensions should the rug have?

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Key Concepts

Uniform Borders in Geometry

Uniform border problems require recognizing that a consistent margin around an object reduces each dimension by twice the border’s width. This insight is essential for accurately forming expressions for the adjusted dimensions of the inner figure (the rug) when compared to the overall space.

Variable Representation in Geometric Context

This concept involves assigning a variable to an unknown quantity, such as the width of the uniform strip or border. By representing the unknown dimension with a variable, one can express the modified dimensions of geometric figures and set up algebraic equations to model real-world scenarios.

Area of a Rectangle

Understanding that the area of a rectangle is found by multiplying its length by its width is fundamental. This principle is used to relate the dimensions of the rug to its area, which is critical when forming an equation based on the problem's constraints.

Formulating and Solving Quadratic Equations

When the geometric relationships are translated into algebraic equations, they often result in quadratic equations. Mastering techniques such as factoring, using the quadratic formula, or completing the square is crucial for solving these equations and determining the unknown quantities.

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