Final answer:
The dimensions of the rectangle, where the length is 3 yards less than twice the width and the area is 27 yd.², are found by setting up a quadratic equation derived from the relationship between length, width, and area. Solving for the width, we find it to be 3 yards, and consequently, the length is also 3 yards, resulting in a square with 3-yard sides.
Step-by-step explanation:
To find the dimensions of the rectangle where the length is 3 yards less than twice the width and the area is 27 yd.², we can set up an equation. Let W represent the width of the rectangle. Therefore, the length is represented as 2W - 3. The area of a rectangle is calculated by multiplying the length and width, leading to our equation:
W × (2W - 3) = 27
Solving for W, we expand and rearrange the equation:
2W² - 3W = 27
Subtracting 27 from both sides gives:
2W² - 3W - 27 = 0
Factoring this quadratic equation, we find:
(2W + 9)(W - 3) = 0
Setting each factor to zero gives us two possible solutions for W:
2W + 9 = 0 —> Not a valid solution since width cannot be negative.
W - 3 = 0 —> W = 3 yards
We find that the width of the rectangle is 3 yards and consequently, the length using the initial expression is 2(3) - 3, which equals 3 yards as well. Therefore, the dimensions of the rectangle are 3 yards by 3 yards.