A rectangle is inscribed in a circle of radius r. If the rectangle has length 2x, write the area of the rectangle as the product of two functions.
To find the area of the rectangle, we need to express it as the product of two functions. Let's start with the diagram of the rectangle inscribed in a circle:
```
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| o | r
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|_______|
```
From the diagram, it's apparent that the diagonal of the rectangle is the diameter of the circle, which is 2r. Thus, we can express the length of the diagonal as 2r.
Since the rectangle is inscribed in the circle, it means that the diagonal of the rectangle is also a diameter of the circle. Now, let's use the Pythagorean theorem to find the length of the other side of the rectangle, which is the height.
Using the Pythagorean theorem, we have:
(diagonal)^2 = (length)^2 + (height)^2
(2r)^2 = (2x)^2 + height^2
4r^2 = 4x^2 + height^2
Simplifying the equation, we get:
height^2 = 4r^2 - 4x^2
Now, we can express the area of the rectangle as the product of two functions:
Area of rectangle = length x height
Area of rectangle = 2x * sqrt(4r^2 - 4x^2)
So, the area of the rectangle can be written as the product of the functions 2x and sqrt(4r^2 - 4x^2).