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To solve the problem presented in the image, we need to derive a mathematical model for the area of a rectangle inscribed in region \( R \), in terms of \( x \), where the boundary of the region is determined by the equation:

\[
x = 9 - y^2
\]

### Understanding the problem:
1. The parabola described by \( x = 9 - y^2 \) defines the boundary of the region \( R \) where the rectangle is inscribed.
2. Points \( P(x, y) \) and \( Q(x, -y) \) represent the top and bottom corners of the rectangle on the parabola.
3. The width of the rectangle is determined by the distance between the points on the parabola, which corresponds to \( 2x \), where \( x = 9 - y^2 \).
4. The height of the rectangle is \( 2y \), given that the height extends from \( -y \) to \( y \).

### Steps to build the model:
1. **Width of the rectangle**: From the equation of the parabola, we know that \( x = 9 - y^2 \). The width of the rectangle will be double the value of \( x \), because the rectangle is symmetric with respect to the \( y \)-axis.
   \[
   \text{Width} = 2x = 2(9 - y^2) = 18 - 2y^2
   \]

2. **Height of the rectangle**: The height of the rectangle is determined by the vertical distance between \( y \) and \( -y \), which is simply:
   \[
   \text{Height} = 2y
   \]

3. **Area of the rectangle**: The area \( A \) of the rectangle is given by the product of its width and height:
   \[
   A = \text{Width} \times \text{Height} = (18 - 2y^2)(2y)
   \]
   Expanding this expression:
   \[
   A = 36y - 4y^3
   \]

### Expressing the area in terms of \( x \):
From the equation \( x = 9 - y^2 \), we can solve for \( y^2 \):
\[
y^2 = 9 - x
\]
Now, substitute \( y^2 = 9 - x \) into the equation for the area:

\[
A = 36y - 4y^3
\]

Substitute \( y = \sqrt{9 - x} \) and \( y^3 = (9 - x)^{3/2} \):

\[
A = 36\sqrt{9 - x} - 4(9 - x)^{3/2}
\]

Thus, the model for the area \( A \) of the rectangle in terms of \( x \) is:

\[
A(x) = 36\sqrt{9 - x} - 4(9 - x)^{3/2}
\]

This represents the area of the inscribed rectangle as a function of \( x \).

---

Would you like a more detailed breakdown of any specific part of the solution?

Here are some follow-up questions you might find helpful:
1. How would you find the maximum area of the rectangle?
2. What happens to the area when \( x = 0 \) or \( x = 9 \)?
3. Can you generalize this problem for a different parabola?
4. How can you confirm if this model is correct for different values of \( x \)?
5. How does the symmetry of the parabola help in simplifying the problem?

**Tip**: To maximize or minimize the area, you could differentiate the area function with respect to \( x \) and solve for critical points.

Obtener el modelo matemático que represente el área de un rectángulo inscrito en la región R, que se muestra en la figura, en términos de x.

Learn how to derive the mathematical model for the area of a rectangle inscribed in the region bounded by the parabola x = 9 - y^2. This problem involves key concepts such as algebra, parabolas, and area calculations, and is suitable for high school and college students.

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