Find the volume and the dimensions of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex on the paraboloid z = 8-x²-2y². a) Find the volume function for this rectangle. b) Maximize the volume of this rectangle. c) What are the dimensions of this largest possible rectangular box?
Find the volume and the dimensions of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex on the paraboloid z = 8-x²-2y². a) Find the volume function for this rectangle. b) Maximize the volume of this rectangle. c) What are the dimensions of this largest possible rectangular box?
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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Please explain well
### Questions
**a)** Find the **volume function** for this rectangle.
**b)** **Maximize** the volume of this rectangle.
**c)** What are the **dimensions** of this largest possible rectangular box?
## Diagram Explanation
The diagram included in the problem statement shows a graphical representation of the paraboloid \( z = 8 - x^2 - 2y^2 \). A rectangular box is inscribed within the first octant, with one of its vertices touching the surface of the paraboloid. The three faces of the box lie on the coordinate planes (the \(xy\)-plane, \(xz\)-plane, and \(yz\)-plane).
The paraboloid surface is depicted in blue, with a wireframe mesh overlay to visualize its 3D shape better. Axes for \(x\), \(y\), and \(z\) are labeled, showing the orientation of the plotted region. The vertex of the rectangular box on the paraboloid and its intersection with the coordinate planes is part of the problem that needs to be solved mathematically.
By solving the problem, you will determine the mathematical function that defines the volume of the box based on its dimensions and maximize this volume to find the optimal dimensions that produce the largest possible rectangular box within the given constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2d62955-1b70-4802-8745-35e1420bbea7%2Fb8c41835-5a35-4d81-ab59-42c2c0cc74a6%2Fkvobpqf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Find the **volume** and the **dimensions** of the largest rectangular box in the **first octant** with three faces in the coordinate planes and **one vertex** on the paraboloid \( z = 8 - x^2 - 2y^2 \).
### Questions
**a)** Find the **volume function** for this rectangle.
**b)** **Maximize** the volume of this rectangle.
**c)** What are the **dimensions** of this largest possible rectangular box?
## Diagram Explanation
The diagram included in the problem statement shows a graphical representation of the paraboloid \( z = 8 - x^2 - 2y^2 \). A rectangular box is inscribed within the first octant, with one of its vertices touching the surface of the paraboloid. The three faces of the box lie on the coordinate planes (the \(xy\)-plane, \(xz\)-plane, and \(yz\)-plane).
The paraboloid surface is depicted in blue, with a wireframe mesh overlay to visualize its 3D shape better. Axes for \(x\), \(y\), and \(z\) are labeled, showing the orientation of the plotted region. The vertex of the rectangular box on the paraboloid and its intersection with the coordinate planes is part of the problem that needs to be solved mathematically.
By solving the problem, you will determine the mathematical function that defines the volume of the box based on its dimensions and maximize this volume to find the optimal dimensions that produce the largest possible rectangular box within the given constraints.
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