Joan wants to buy a rug for a room that is 8 feet by 15 feet. She wants to leave a uniform strip of floor around the rug. She can afford 60 square feet of carpeting. What dimensions should the rug have? Let \( x \) represent the width of the uniform strip of floor around the rug. Set up an equation that can be used to solve for \( x \).

To solve for the dimensions of the rug that Joan wants to buy, we can follow these steps: 1. **Identify the dimensions of the room**: The room is 8 feet by 15 feet. 2. **Define the variable**: Let \( x \) represent the width of the uniform strip of floor around the rug. 3. **Determine the dimensions of the rug**: - The length of the rug will be \( 15 - 2x \) (subtracting \( x \) from both ends of the length). - The width of the rug will be \( 8 - 2x \) (subtracting \( x \) from both ends of the width). 4. **Calculate the area of the rug**: The area \( A \) of the rug can be expressed as: \[ A = (15 - 2x)(8 - 2x) \] 5. **Set up the equation**: Since Joan can afford 60 square feet of carpeting, we set the area of the rug equal to 60: \[ (15 - 2x)(8 - 2x) = 60 \] Now we have the equation that can be used to solve for \( x \): \[ (15 - 2x)(8 - 2x) = 60 \] Next, we can expand this equation and solve for \( x \). Let's proceed with that. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(\left(15-2x\right)\left(8-2x\right)=60\) - step1: Expand the expression: \(120-46x+4x^{2}=60\) - step2: Move the expression to the left side: \(60-46x+4x^{2}=0\) - step3: Factor the expression: \(2\left(10-x\right)\left(3-2x\right)=0\) - step4: Divide the terms: \(\left(10-x\right)\left(3-2x\right)=0\) - step5: Separate into possible cases: \(\begin{align}&10-x=0\\&3-2x=0\end{align}\) - step6: Solve the equation: \(\begin{align}&x=10\\&x=\frac{3}{2}\end{align}\) - step7: Rewrite: \(x_{1}=\frac{3}{2},x_{2}=10\) The solutions for \( x \) from the equation \((15 - 2x)(8 - 2x) = 60\) are: 1. \( x_1 = \frac{3}{2} = 1.5 \) 2. \( x_2 = 10 \) However, \( x = 10 \) is not a feasible solution because it would imply that the rug has negative dimensions (since the room is only 8 feet wide). Therefore, the only valid solution is: \[ x = 1.5 \] ### Conclusion: The width of the uniform strip of floor around the rug should be \( 1.5 \) feet. Now, we can find the dimensions of the rug: - Length of the rug: \( 15 - 2(1.5) = 15 - 3 = 12 \) feet - Width of the rug: \( 8 - 2(1.5) = 8 - 3 = 5 \) feet Thus, the dimensions of the rug should be **12 feet by 5 feet**.