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We are given that a rectangle is inscribed in a circle of radius 5, and we need to express the area of the rectangle as a function of \(x\).

### Step 1: Equation of the Circle
The circle has a radius of 5, so its equation is:
\[
x^2 + y^2 = 25
\]
The point \(P(x, y)\) is on the circle and represents one of the vertices of the rectangle in the first quadrant.

### Step 2: Relationship Between \(x\) and \(y\)
From the equation of the circle, we can express \(y\) in terms of \(x\):
\[
y = \sqrt{25 - x^2}
\]

### Step 3: Dimensions of the Rectangle
Since the rectangle is symmetric with respect to both the x-axis and y-axis, its width is \(2x\) (because the vertex at \(P(x, y)\) is at \(x\), and the other vertex is at \(-x\)) and its height is \(2y\) (similarly, the height extends from \(y\) to \(-y\)).

- Width = \(2x\)
- Height = \(2y = 2\sqrt{25 - x^2}\)

### Step 4: Area of the Rectangle
The area \(A(x)\) of the rectangle is the product of its width and height:
\[
A(x) = 2x \cdot 2\sqrt{25 - x^2}
\]
Simplifying:
\[
A(x) = 4x\sqrt{25 - x^2}
\]

Thus, the area of the rectangle as a function of \(x\) is:
\[
A(x) = 4x\sqrt{25 - x^2}
\]

Let me know if you'd like further clarification!

Here are 5 related questions to expand your understanding:
1. How does the area of the rectangle change as \(x\) approaches 0?
2. What is the maximum possible area for the rectangle, and at what \(x\) does it occur?
3. How would the equation change if the radius of the circle were different?
4. What are the coordinates of the vertex \(P\) when the rectangle has its maximum area?
5. How does the symmetry of the problem help in simplifying the calculations?

**Tip:** To find the maximum area of the rectangle, you can take the derivative of \(A(x)\) and find its critical points.

A rectangle is inscribed in a circle of radius 5. Let P = (x, y) be the point in quadrant I that is a vertex of the rectangle and is on the circle. Express the area A of the rectangle as a function of x.

Learn how to express the area of a rectangle inscribed in a circle of radius 5 as a function of x. This problem covers geometry, the equation of a circle, and function modeling. Suitable for students in Grades 10-12.

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