Final answer:
To find the dimensions that will create a rectangle using all the given rope, we can use the formula for the perimeter of a rectangle and the concept of maximum or minimum value of a quadratic equation. Using these concepts, we find that the dimensions of the rectangle are 35 meters by 35 meters.
Step-by-step explanation:
Ashley is given 140 meters of rope and 4 stakes to mark off a rectangular area. In order to determine the dimensions that will create a rectangle using all the rope Ashley has, we need to consider the formula for the perimeter of a rectangle, which is 2L + 2W, where L represents the length and W represents the width. Since the perimeter is equal to 140 meters, we can set up the equation 2L + 2W = 140. We can rearrange this equation to solve for L in terms of W, L = 70 - W. Substituting this equation into the area formula A = LW, we get A = W(70 - W), which simplifies to A = 70W - W^2. To find the dimensions that will maximize the area, we can use the concept of maximum or minimum value of a quadratic equation, which occurs at the vertex. The x-coordinate of the vertex is given by the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c. In our case, a = -1 (since the coefficient of W^2 is -1), b = 70, and c = 0. Plugging these values into the formula, we get W = -70/(-2) = 35. Therefore, the width of the rectangle is 35 meters and the length is 70 - 35 = 35 meters as well. So, the dimensions that will create a rectangle using all the rope Ashley has with her are 35 meters by 35 meters.