Final answer:
To find the dimensions of the rug, we need to subtract the width of the uniform strip of floor from the room's width and length. By solving the resulting quadratic equation, we find that the dimensions of the rug should be 4 feet by 13 feet.
Step-by-step explanation:
To find the dimensions of the rug, we need to subtract the width of the uniform strip of floor from the room's width and length. Let's call the width of the strip 'w'. The width of the rug would then be 22 feet - 2w and the length would be 31 feet - 2w. Now we can form an equation based on the area of the rug: (22 - 2w)(31 - 2w) = 322. Solving this equation will give us the dimensions of the rug.
We can start by expanding the equation: 682 - 54w + 4w^2 = 322. Rearranging the terms, we get 4w^2 - 54w + 360 = 0. To solve this quadratic equation, we can factor it: (2w - 18)(2w - 10) = 0. By setting each factor equal to zero, we find two potential values for 'w': w = 9 and w = 5.
However, we need to find the dimensions of the rug, not the width of the strip. So we substitute w = 9 into the equation: width = 22 - 2w = 22 - 2(9) = 22 - 18 = 4 feet. Similarly, length = 31 - 2w = 31 - 2(9) = 31 - 18 = 13 feet. Therefore, the dimensions of the rug should be 4 feet by 13 feet, which means the correct answer is Option D) 9 feet by 14 feet.