A rectangle has one side on the x-axis and two vertices on the curve
y=6/{1+x^2}.
Find the vertices of the rectangle with maximum area.
Vertices =
To find the vertices of the rectangle with maximum area, we need to find the maximum value of the area function and then calculate the coordinates of the vertices corresponding to that maximum value.
Let's start by finding the area function of the rectangle. The area of a rectangle is calculated by multiplying its length by its width. In this case, the length of the rectangle will be the distance between the two vertices on the curve, and the width will be the length of the side on the x-axis.
Let's assume the coordinates of the two vertices on the curve are (x1, y1) and (x2, y2), and the length of the rectangle is L and the width is W.
Since one side of the rectangle is on the x-axis, the width is determined by the difference in x-coordinates of the two vertices: W = |x2 - x1|.
The length of the rectangle can be calculated using the distance formula: L = sqrt((x2 - x1)^2 + (y2 - y1)^2).
Now we can calculate the area of the rectangle:
Area = Length * Width = L * W = sqrt((x2 - x1)^2 + (y2 - y1)^2) * |x2 - x1|.
To find the maximum area, we need to differentiate the area function with respect to x. Let's call the area function f(x) and find its derivative f'(x).
f(x) = sqrt((x2 - x1)^2 + (y2 - y1)^2) * |x2 - x1|.
f'(x) = 0, where f'(x) is the derivative of f(x).
Differentiating f(x) with respect to x and setting it equal to zero will help us find the x-values of the vertices of the rectangle with maximum area.
Once we find the x-values, we can calculate the corresponding y-values by substituting them back into the original equation y = 6/(1 + x^2).
Finally, we can obtain the coordinates of the vertices of the rectangle with maximum area by using the x and y values we found.