A rectangle has one corner in quadrant I on the graph of y = 9 − x^2,another corner at the origin, and corners on both the positive y-axis and the positive x-axis. As the corner on

y = 9 − x2 changes, a variety of rectangles are obtained.

Express the area A of the rectangles as a function of x.

clearly, each rectangle with upper-right corner at (x,y) has area

a = xy

Since y = 9-x^2, the area is just

a = x(9-x^2)

Not sure what you need to do with that information. The question is a bit vague.

To express the area A of the rectangles as a function of x, we need to find the length and width of the rectangle in terms of x.

Given that one corner of the rectangle is at the origin, we know that the width of the rectangle is equal to x.

To find the length of the rectangle, we need to determine the y-coordinate of the point where the other corner of the rectangle lies on the graph of y = 9 - x^2.

Substituting x into the equation, we can find the y-coordinate as follows:

y = 9 - x^2

Therefore, the length of the rectangle is equal to 9 - x^2.

The area A of a rectangle is given by the formula A = length × width. Substituting the expressions we found earlier, we have:

A = (9 - x^2) × x

Simplifying this equation, we get the area A as a function of x:

A = 9x - x^3

To express the area A of the rectangles as a function of x, we need to determine the length of the sides of the rectangle.

Let's consider the corner on the graph of y = 9 - x^2. The x-coordinate of this corner will be the length of the base of the rectangle, and the y-coordinate will be the length of the height of the rectangle.

The corner on the positive y-axis will have coordinates (0, y), where y is the length of the base of the rectangle.

The corner on the positive x-axis will have coordinates (x, 0), where x is the length of the height of the rectangle.

So, the length of the base will be the x-coordinate of the corner on the graph: x = x.

And the length of the height will be the y-coordinate of the corner on the graph: y = 9 - x^2.

Therefore, the area A of the rectangle can be expressed as the product of the base and the height:

A = x * (9 - x^2)

Now, the expression A = x * (9 - x^2) represents the area of the rectangles as a function of x.