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?

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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.
The rug should be 19 feet wide and 27 feet long.

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?

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