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We are tasked with expressing the area \( A(x) \) of a rectangle inscribed in a circle of radius 6 as a function of \( x \), where \( P = (x, y) \) is a vertex of the rectangle.

### Step-by-Step Breakdown:

1. **Equation of the Circle**:  
   The equation of a circle with center at the origin and radius \( 6 \) is:
   \[
   x^2 + y^2 = 36
   \]
   This equation will help us relate \( x \) and \( y \) for any point \( P(x, y) \) on the circle.

2. **Dimensions of the Rectangle**:  
   The rectangle is inscribed in the circle. Given the symmetry of the figure, if \( P(x, y) \) is a vertex of the rectangle in Quadrant I, then the opposite vertex of the rectangle would be at \( (-x, -y) \).

   - The **width** of the rectangle (along the \( x \)-axis) would be \( 2x \).
   - The **height** of the rectangle (along the \( y \)-axis) would be \( 2y \).

3. **Area of the Rectangle**:  
   The area \( A(x) \) of the rectangle is the product of its width and height:
   \[
   A(x) = \text{width} \times \text{height} = 2x \times 2y = 4xy
   \]

4. **Substitute \( y \) in Terms of \( x \)**:  
   From the equation of the circle, we can solve for \( y \) in terms of \( x \):
   \[
   y = \sqrt{36 - x^2}
   \]
   Thus, the area function becomes:
   \[
   A(x) = 4x \cdot \sqrt{36 - x^2}
   \]

### Final Answer:
\[
A(x) = 4x \sqrt{36 - x^2}
\]

This is the expression for the area of the rectangle as a function of \( x \).

Would you like to explore any further details on this? Here are five related questions:

1. How can we determine the maximum possible area of the rectangle?
2. What is the domain of the function \( A(x) \)?
3. How does the graph of \( A(x) \) look, and where does it reach its maximum?
4. Can we solve for \( x \) when the area is a given value?
5. How can we extend this problem to a general circle of radius \( r \)?

**Tip**: Always double-check the domain of your function to ensure it makes sense in the context of the problem. For example, \( x \) must be between \( -6 \) and \( 6 \), inclusive, because the radius of the circle is 6.

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 6 as a function of x. This problem involves key concepts in geometry, including the equation of a circle and the Pythagorean theorem. Suitable for students in Grades 10-12.

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