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A rectangle is inscribed with its base on the x-axis and its upper corners on the semi-ellipse defined by x² + 23y² = 25 in Quadrants I and II. What is the greatest possible area?

A rectangle is inscribed with its base on the x-axis and its upper corners on the semi-ellipse defined by x² + 23y² = 25 in Quadrants I and II. What is the greatest possible area?

Intermediate Algebra
Intermediate Algebra
19th Edition
ISBN:9780998625720
Author:Lynn Marecek
Publisher:Lynn Marecek
Chapter11: Conics
Section11.2: Parabolas
Problem 97E: Write the equation of a parabola that opens up or down in standard form and the equation of a...
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The maximum area =  ?

(use calculus)

The image shows a mathematical problem involving a semi-ellipse and an inscribed rectangle. The ellipse is defined by the equation \(x^2 + 23y^2 = 25\). The rectangle is set with its base along the x-axis, and its top corners are positioned on the semi-ellipse within Quadrants I and II. The problem asks for the greatest possible area of this inscribed rectangle. ### Diagram Description: - **Semi-Ellipse:** The semi-ellipse is shown with its major axis along the x-axis and the top part of the ellipse visible in the positive y-plane. - **Rectangle:** The rectangle is inscribed such that its base lies on the x-axis. Its height stretches upward to meet the semi-ellipse. The rectangle is symmetrically placed about the vertical axis of the ellipse. ### Key Features: - The base of the rectangle is horizontal and the width extends equally on both sides of the y-axis. - The upper corners of the rectangle just touch the boundary of the semi-ellipse, indicating that the width and height are limited by the ellipse equation. This setup typically involves using calculus—specifically optimization techniques—to find the maximum area of the rectangle inscribed within these constraints.
Transcribed Image Text:The image shows a mathematical problem involving a semi-ellipse and an inscribed rectangle. The ellipse is defined by the equation \(x^2 + 23y^2 = 25\). The rectangle is set with its base along the x-axis, and its top corners are positioned on the semi-ellipse within Quadrants I and II. The problem asks for the greatest possible area of this inscribed rectangle. ### Diagram Description: - **Semi-Ellipse:** The semi-ellipse is shown with its major axis along the x-axis and the top part of the ellipse visible in the positive y-plane. - **Rectangle:** The rectangle is inscribed such that its base lies on the x-axis. Its height stretches upward to meet the semi-ellipse. The rectangle is symmetrically placed about the vertical axis of the ellipse. ### Key Features: - The base of the rectangle is horizontal and the width extends equally on both sides of the y-axis. - The upper corners of the rectangle just touch the boundary of the semi-ellipse, indicating that the width and height are limited by the ellipse equation. This setup typically involves using calculus—specifically optimization techniques—to find the maximum area of the rectangle inscribed within these constraints.
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