**Title: Problem 4.3/4.7 - Rectangle Inscribed in an Ellipse**A rectangle is inscribed in the ellipse characterized by the equation \(\frac{x^2}{4} + \frac{y^2}{1} = 1\).**Tasks:**1. **Find the minimum possible area of the rectangle.**2. **Find the maximum possible area of the rectangle.**3. **Find the maximum possible perimeter of the rectangle.****Solution Steps:**The problem involves finding specific attributes of a rectangle inscribed in an ellipse with a given equation. The goal is to determine extreme values related to area and perimeter.**Solution Diagram and Annotations:**- On the right, there is a sketch of the ellipse and the rectangle inscribed within it. The ellipse is centered at the origin, extending horizontally to 2 units and vertically to 1 unit. The rectangle is shaded inside and aligned symmetrically with respect to both axes.**Mathematical Analysis:**- **Area Calculation:** - The area of the rectangle \(A\) can be expressed as a function of \(x\): \(A = 2x \cdot 2y\), where \(y = \sqrt{1 - \frac{x^2}{4}}\). - Substituting \(y\), the area \(A = 2x \times 2\sqrt{1 - \frac{x^2}{4}}\).- **Perimeter Calculation:** - Not directly solved here, but involves using the expression for the perimeter \(P = 2(l + w)\), where \(l\) and \(w\) are the side lengths related to \(x\) and \(y\).- **Further Annotations:** - There are additional handwritten notes involving equations and calculations attempting to find extreme values using derivatives or completing the square. - Computational steps involve manipulating the equation \(D^2 = (1 - \frac{x^2}{4})\).This analysis allows a deeper understanding of the geometric properties of shapes inscribed within conic sections, specifically focusing on the ellipse and optimization problems related to area and perimeter.

Given equation of the ellipse is x24+y2=1 The general equation of an ellipse is x2a2+y2b2=1 Then…

1. 4.3/4.7: A rectangle is inscribed in the ellipse+y² a) Find the minimum possible area of the rectangle. (b) Find the maximum possible area of the rectangle. (c) Find the maximum possible perimeter of the rectangle.

1. 4.3/4.7: A rectangle is inscribed in the ellipse+y² a) Find the minimum possible area of the rectangle. (b) Find the maximum possible area of the rectangle. (c) Find the maximum possible perimeter of the rectangle.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A: The given problem is solved below in detail.

Question

**Title: Problem 4.3/4.7 - Rectangle Inscribed in an Ellipse**

A rectangle is inscribed in the ellipse characterized by the equation \(\frac{x^2}{4} + \frac{y^2}{1} = 1\).

**Tasks:**

1. **Find the minimum possible area of the rectangle.**

2. **Find the maximum possible area of the rectangle.**

3. **Find the maximum possible perimeter of the rectangle.**

**Solution Steps:**

The problem involves finding specific attributes of a rectangle inscribed in an ellipse with a given equation. The goal is to determine extreme values related to area and perimeter.

**Solution Diagram and Annotations:**

- On the right, there is a sketch of the ellipse and the rectangle inscribed within it. The ellipse is centered at the origin, extending horizontally to 2 units and vertically to 1 unit. The rectangle is shaded inside and aligned symmetrically with respect to both axes.

**Mathematical Analysis:**

- **Area Calculation:**
- The area of the rectangle \(A\) can be expressed as a function of \(x\):
\(A = 2x \cdot 2y\), where \(y = \sqrt{1 - \frac{x^2}{4}}\).
- Substituting \(y\), the area \(A = 2x \times 2\sqrt{1 - \frac{x^2}{4}}\).

- **Perimeter Calculation:**
- Not directly solved here, but involves using the expression for the perimeter \(P = 2(l + w)\), where \(l\) and \(w\) are the side lengths related to \(x\) and \(y\).

- **Further Annotations:**
- There are additional handwritten notes involving equations and calculations attempting to find extreme values using derivatives or completing the square.
- Computational steps involve manipulating the equation \(D^2 = (1 - \frac{x^2}{4})\).

This analysis allows a deeper understanding of the geometric properties of shapes inscribed within conic sections, specifically focusing on the ellipse and optimization problems related to area and perimeter.

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4.3/4.7: A rectangle is inscribed in the ellipse ±y ²
a) Find the minimum possible area of the rectangle:
(b) Find the maximum possible area of the rectangle.
(c) Find the maximum possible perimeter of the rectangle.
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