Use Lagrange multipliers to find the maximum area S of a rectangle inscribed in the ellipse x² 1² 49 9 (-x, y) (-x, -y) (x, y) (x, -y) Give your answer as a whole or exact number.)
Use Lagrange multipliers to find the maximum area S of a rectangle inscribed in the ellipse x² 1² 49 9 (-x, y) (-x, -y) (x, y) (x, -y) Give your answer as a whole or exact number.)
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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Question
![**Problem:**
Use Lagrange multipliers to find the maximum area \( S \) of a rectangle inscribed in the ellipse
\[
\frac{x^2}{9} + \frac{y^2}{49} = 1
\]
**Diagram Explanation:**
The image includes a diagram of an ellipse centered at the origin with axes labeled \( x \) and \( y \). The ellipse intersects the \( x \)-axis at \( (\pm 3, 0) \) and the \( y \)-axis at \( (0, \pm 7) \). A rectangle is inscribed within the ellipse, with its sides parallel to the coordinate axes. The vertices of the rectangle are denoted as \( (x, y) \), \( (-x, y) \), \( (x, -y) \), and \( (-x, -y) \).
**Task:**
(Give your answer as a whole or exact number.)
\[
S = \_\_\_\_\_\_\_\_
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53419bf8-add0-48cb-b87d-efce75dfc052%2Fa9091695-c0b3-4314-8415-e5b05d25e19b%2Fdscimp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Use Lagrange multipliers to find the maximum area \( S \) of a rectangle inscribed in the ellipse
\[
\frac{x^2}{9} + \frac{y^2}{49} = 1
\]
**Diagram Explanation:**
The image includes a diagram of an ellipse centered at the origin with axes labeled \( x \) and \( y \). The ellipse intersects the \( x \)-axis at \( (\pm 3, 0) \) and the \( y \)-axis at \( (0, \pm 7) \). A rectangle is inscribed within the ellipse, with its sides parallel to the coordinate axes. The vertices of the rectangle are denoted as \( (x, y) \), \( (-x, y) \), \( (x, -y) \), and \( (-x, -y) \).
**Task:**
(Give your answer as a whole or exact number.)
\[
S = \_\_\_\_\_\_\_\_
\]
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