Final answer:
To find the length and width of the rectangular carpet, we can set up a quadratic equation based on the given information and solve it using the quadratic formula. The length of the carpet is approximately 18.2 yards, and the width is approximately 9.2 yards.
Step-by-step explanation:
To solve this problem, we can set up a system of equations based on the given information. Let the length of the carpet be represented by 'x' and the width be represented by 'x-9'. We know that the area of a rectangle is found by multiplying the length and width, so we can set up the equation x(x-9) = 200. Simplifying this equation gives us x^2 - 9x - 200 = 0. We can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Once we find the value of x, we can substitute it back into the expression x-9 to find the width. Let's solve the equation using the quadratic formula: x = (-(-9) ± √((-9)^2 - 4(1)(-200)))/(2(1)). Simplifying this gives us x = (9 ± √(81+800))/2, x = (9 ± √881)/2. Taking the positive value, x = (9 + √881)/2 ≈ 18.2. So, the length of the carpet is approximately 18.2 yards. Substituting this value back into x-9, we find the width to be approximately 9.2 yards.