A rectangle with its base on the x-axis is to be inscribed under the graph of y=2-x^2. Find the height of the rectangle if the area is the largest possible area. (Round to the nearest hundreth)
Let the point of contact of the rectangle in the first quadrant be (x,y)
then the base of the rectangle is 2x and the height is y
Area = 2xy
= 2x(2-x^2)
= 4x - 2x^3
d(Area)/dx = 4 - 6x^2
= 0 for a max of Area
6x^2 = 4
x^2 = 4/6
x = ±2/6
then height = y = 2 - 4/6 = 4/3
= 1.33 to the nearest hundreth
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!
To find the largest possible area of the rectangle inscribed under the graph of y=2-x^2, we need to determine the dimensions (base and height) that will maximize the area.
Let's start by visualizing the problem. The graph of y=2-x^2 is a downward-opening parabola symmetric about the y-axis. When we draw a rectangle under this graph with its base on the x-axis, we find that the rectangle will have its four corners touching the curve of the parabola.
To find the dimensions that maximize the area, we need to determine the x-values of the corners of the rectangle. To do this, we can set y=0 in the equation of the parabola:
0 = 2 - x^2
Solving for x, we have:
x^2 = 2
Taking the square root of both sides, we get:
x = ±√2
Since the rectangle has its base on the x-axis, the x-values of the corners are ±√2.
Now, let's calculate the height of the rectangle. Since the top corners of the rectangle touch the parabola, we need to find the corresponding y-values. We substitute the x-values ±√2 into the equation of the parabola:
When x = √2:
y = 2 - (√2)^2
y = 2 - 2
y = 0
When x = -√2:
y = 2 - (-√2)^2
y = 2 - 2
y = 0
Both corresponding y-values are zero, meaning the height of the rectangle is 0.
Therefore, the height of the rectangle, when the area is maximized, is 0.
To find the maximum area of the rectangle, we need to determine the dimensions that will maximize it. In this case, the base of the rectangle will be along the x-axis, and the height will be along the y-axis.
First, let's draw the graph of y = 2 - x^2 to visualize the situation.
The graph is a downward-opening parabola symmetric to the y-axis. The vertex of the parabola is located at (0, 2), and the curve intersects the x-axis at two points: (-√2, 0) and (√2, 0).
We want to inscribe a rectangle with its base on the x-axis under this curve. This means the base length of the rectangle will be the distance between the x-intercepts of the parabola, which is 2√2.
To find the maximum area, we need to determine the height of the rectangle. The height will be the y-coordinate of a point on the parabola between the x-intercepts.
To find this point, we can take the derivative of the equation y = 2 - x^2 with respect to x, set it equal to zero, and solve for x. This will give us the x-coordinate of the point where the slope of the curve is zero, which corresponds to the maximum area.
dy/dx = -2x
Setting -2x = 0, we find that x = 0.
So, the x-coordinate of the maximum area point is 0. To find the corresponding height (y-coordinate), we substitute this value back into the equation y = 2 - x^2:
y = 2 - (0)^2
y = 2
Therefore, the height of the rectangle is 2.
Finally, we can calculate the area of the rectangle by multiplying the base length by the height:
Area = base * height = 2√2 * 2 = 4√2
Rounding 4√2 to the nearest hundredth, the height of the rectangle with the largest possible area is approximately 5.66.