A rectangle is inscribed with it's base on the x-axis and it's upper corners on the parabola y=4-x^2. What are the dimensions of such a rectangle with the greatest possible area?
Width=
Height=
the area is, as always, length * height
Let the base be of size 2x, with its center at (0,0) and the corners at ±x
a = 2x(4-x^2)
da/dx = 8-6x^2
so the max area is 32/(3√3) when x=2/√3
Thus the rectangle's dimensions are 4/√3 by 8/3
To find the dimensions of the rectangle with the greatest possible area, we need to determine the coordinates of the rectangle's corners on the parabola y=4-x^2.
Let's start by visualizing the problem. We have a rectangle inscribed in the parabola, with its base on the x-axis. The height of the rectangle will be the distance from the x-axis to the y-coordinate on the parabola, and the width will be the distance between the x-coordinates of the two corners.
To maximize the area of the rectangle, the width needs to be as large as possible. This means both corners of the rectangle should be on the x-axis. Since the parabola is symmetric with respect to the y-axis, the x-coordinates of the two corners will have the same absolute value but opposite signs. Let's call them x and -x.
Now, we need to find the y-coordinates of these points on the parabola. Plugging in the values of x and -x into y=4-x^2, we can calculate the y-coordinates.
For x, we have:
y = 4 - x^2
y = 4 - (-x)^2
y = 4 - x^2
So, the y-coordinate for the corner at (x, y) is y = 4 - x^2.
Finally, we can find the dimensions of the rectangle. The height is the absolute value of the y-coordinate on the parabola:
Height = |y| = |4 - x^2|.
The width is the distance between the x-coordinates of the two corners:
Width = 2x, since one corner is at x and the other is at -x.
To find the dimensions that result in the greatest possible area, we need to maximize the area function, which is the product of the width and height:
Area = Width * Height
= 2x * |4 - x^2|.
To find the maximum area, we can take the derivative of the area function with respect to x, set it equal to zero, and solve for x. This will give us the x-coordinate of the corner that maximizes the area.