Final answer:
To find the dimensions of the rectangle of largest area, we need to consider the parabola and the x-axis. The dimensions are determined by the x-coordinate and the y-coordinate of the two vertices above the x-axis. By finding the maximum value of the area formula, we can determine the dimension of the rectangle of largest area.
Step-by-step explanation:
To find the dimensions of the rectangle of largest area, we need to consider the parabola and the x-axis. Let's assume the coordinates of the two vertices above the x-axis are (x, y) and (-x, y), where y is the value of the parabola at x.
The area of a rectangle is given by length x width. In this case, the length is 2x (twice the x-coordinate) and the width is y (the y-coordinate).
Therefore, the area of the rectangle is A = 2x * y.
To maximize the area, we need to find the maximum value of A. This can be done by finding the maximum value of y for a given x.
Since the other two vertices lie on the parabola, we have y = x^2. Substituting this into the area formula, we get A = 2x * x^2 = 2x^3.
To find the maximum value of A, we can take the derivative of A with respect to x and set it equal to 0. Taking the derivative, we get dA/dx = 6x^2.
Setting this equal to 0, we find x = 0 or x = ±√(3/2).
Since we want the dimension of the rectangle, we take the positive value of x, which is √(3/2).
Therefore, the dimension of the rectangle of largest area is approximately 1.22 units.