Cynthia Besch wants to buy a rug for a room that is 25 ft wide and 33 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 513 square feet of carpeting. What dimensions should the rug have?

To determine the dimensions of the rug Cynthia Besch should buy, follow these steps: 1. **Define the Variables:** - Let \( x \) be the uniform width of the strip around the rug. - The room dimensions are 25 ft (width) and 33 ft (length). 2. **Express Rug Dimensions:** - **Width of the rug:** \( 25 - 2x \) ft - **Length of the rug:** \( 33 - 2x \) ft 3. **Set Up the Area Equation:** \[ (25 - 2x)(33 - 2x) = 513 \text{ square feet} \] 4. **Expand and Simplify:** \[ 25 \times 33 - 50x - 66x + 4x^2 = 513 \] \[ 825 - 116x + 4x^2 = 513 \] \[ 4x^2 - 116x + 312 = 0 \] \[ x^2 - 29x + 78 = 0 \quad (\text{Divide all terms by 4}) \] 5. **Solve the Quadratic Equation:** \[ x = \frac{29 \pm \sqrt{29^2 - 4 \times 1 \times 78}}{2} \] \[ x = \frac{29 \pm \sqrt{841 - 312}}{2} \] \[ x = \frac{29 \pm \sqrt{529}}{2} \] \[ x = \frac{29 \pm 23}{2} \] - **Possible Solutions:** \( x = 26 \) or \( x = 3 \) - **Feasible Solution:** \( x = 3 \) (since \( x = 26 \) would make the rug dimensions negative) 6. **Calculate Rug Dimensions:** - **Width:** \( 25 - 2(3) = 19 \) ft - **Length:** \( 33 - 2(3) = 27 \) ft 7. **Verify the Area:** \[ 19 \times 27 = 513 \text{ square feet} \] **Conclusion:** The rug should measure **19 feet in width** and **27 feet in length** to fit the room with a uniform strip around it. **Answer:** A rug 19 ft wide by 27 ft long will provide a uniform strip around the room.