Well, well, well! Looks like we have a mathematical challenge here. Let's dive right in!
To find the maximum area of the rectangle, we need to maximize its height. And lucky for us, the base of the rectangle is on the good ol' x-axis.
Now, the area of the rectangle is given by A = base * height. Since the base is along the x-axis, the length of the base will be equal to the length of the interval on the x-axis over which the rectangle is inscribed.
So, our task is to find that interval on the x-axis that will yield the largest possible area. To do that, we need to identify where the graph of y = 2 - x^2 intersects the x-axis.
To find the x-intercepts, we set y = 0 and solve for x:
0 = 2 - x^2
x^2 = 2
x = ±√2
Since we're dealing with geometry, we take the positive root:
x = √2
So, the interval on the x-axis over which the rectangle is inscribed is [0, √2]. The length of the base is equal to √2.
Now, to find the height, we need to determine the corresponding y-values on the graph of y = 2 - x^2 for x = 0 and x = √2.
When x = 0:
y = 2 - (0)^2
y = 2
When x = √2:
y = 2 - (√2)^2
y = 2 - 2
y = 0
So, the height of the rectangle is the difference between these two y-values:
height = 2 - 0
height = 2
And there you have it, my mathematically inclined friend! The height of the rectangle, to the nearest hundredth, is 2. So, go forth and create rectangles with great heights!