https://r2.erweima.ai/mathbot/image/1726631276729fm4fz67x.jpg

To solve this problem, we need to find a mathematical model representing the area of the rectangle \( R \), which is inscribed in a region bounded by the curve \( x = 9 - y^2 \), in terms of \( x \).

### 1. Equation of the Curve:
The given curve is a parabola defined by:
\[
x = 9 - y^2
\]
Solving for \( y \), we get:
\[
y = \pm \sqrt{9 - x}
\]
This represents the upper and lower boundaries of the region, with \( P(x, y) \) on the upper boundary and \( Q(x, -y) \) on the lower boundary.

### 2. Width of the Rectangle:
The width of the rectangle is the horizontal distance between the points \( P(x, y) \) and \( Q(x, -y) \), which is the distance between \( x = 0 \) and \( x = 9 - y^2 \).

### 3. Height of the Rectangle:
The height of the rectangle is the vertical distance between \( y = \sqrt{9 - x} \) and \( y = -\sqrt{9 - x} \). Therefore, the height is:
\[
\text{Height} = 2y = 2\sqrt{9 - x}
\]

### 4. Area of the Rectangle:
The area \( A(x) \) of the rectangle is the product of the width and the height. Since the width of the rectangle is \( x \) (the distance from the origin along the \( x \)-axis), and the height is \( 2\sqrt{9 - x} \), the area is given by:
\[
A(x) = x \cdot 2\sqrt{9 - x}
\]
Simplifying:
\[
A(x) = 2x\sqrt{9 - x}
\]

Thus, the mathematical model for the area of the rectangle in terms of \( x \) is:
\[
A(x) = 2x\sqrt{9 - x}
\]

Would you like further details on any part of this solution? Here are some related questions:

1. How do you find the critical points of the area function \( A(x) \)?
2. What is the domain of the area function \( A(x) \)?
3. How can you maximize the area of the rectangle?
4. How does the symmetry of the parabola affect the dimensions of the rectangle?
5. What would the effect be if the parabola equation were modified?

**Tip**: When dealing with geometry problems, identifying symmetry in the figure can often simplify calculations and help you understand the problem better.

Obtain the mathematical model that represents the area of a rectangle inscribed in region R, shown in the figure, in terms of x.

This problem involves finding the area of a rectangle inscribed within a parabolic region defined by the equation x = 9 - y^2. The solution includes modeling the area of the rectangle in terms of x using geometric properties and functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation