Cynthia Besch wants to buy a rug for a room that is 23 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 330 square feet of carpeting. What dimensions should the rug have?

To find the dimensions of the rug that Cynthia Besch can buy, we need to set up an equation based on the area of the room and the area of the rug. 1. **Calculate the area of the room:** \[ \text{Area of the room} = \text{width} \times \text{length} = 23 \, \text{ft} \times 30 \, \text{ft} = 690 \, \text{sq ft} \] 2. **Let \( x \) be the width of the uniform strip of floor around the rug.** Then, the dimensions of the rug will be: - Width of the rug: \( 23 - 2x \) - Length of the rug: \( 30 - 2x \) 3. **Calculate the area of the rug:** \[ \text{Area of the rug} = (23 - 2x)(30 - 2x) \] 4. **Set up the equation for the area of the rug:** Since Cynthia can afford 330 square feet of carpeting, we have: \[ (23 - 2x)(30 - 2x) = 330 \] 5. **Expand the left side:** \[ 23 \times 30 - 2x \times 30 - 2x \times 23 + 4x^2 = 330 \] \[ 690 - 60x - 46x + 4x^2 = 330 \] \[ 690 - 106x + 4x^2 = 330 \] 6. **Rearrange the equation:** \[ 4x^2 - 106x + 690 - 330 = 0 \] \[ 4x^2 - 106x + 360 = 0 \] 7. **Divide the entire equation by 2 to simplify:** \[ 2x^2 - 53x + 180 = 0 \] 8. **Use the quadratic formula to solve for \( x \):** The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 2 \), \( b = -53 \), and \( c = 180 \). Calculate the discriminant: \[ b^2 - 4ac = (-53)^2 - 4 \times 2 \times 180 = 2809 - 1440 = 1369 \] Now, calculate \( x \): \[ x = \frac{53 \pm \sqrt{1369}}{4} \] \[ \sqrt{1369} = 37 \] \[ x = \frac{53 \pm 37}{4} \] This gives us two possible solutions for \( x \): \[ x = \frac{90}{4} = 22.5 \quad \text{(not valid, as it exceeds half the width/length)} \] \[ x = \frac{16}{4} = 4 \] 9. **Calculate the dimensions of the rug:** If \( x = 4 \): - Width of the rug: \( 23 - 2(4) = 23 - 8 = 15 \, \text{ft} \) - Length of the rug: \( 30 - 2(4) = 30 - 8 = 22 \, \text{ft} \) Thus, the dimensions of the rug should be **15 ft wide and 22 ft long**.