### Volume of the Largest Rectangular Box in the First Octant**Problem Statement:**Determine the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, one vertex at the origin, and its opposite vertex in the plane $ x + 3y + 6z = 36 $.**Explanation:**To solve this problem, you need to use calculus, specifically optimization techniques. Here, the goal is to maximize the volume of the rectangular box under given constraints. The constraints are:- The box is in the first octant, meaning $ x, y, z \geq 0 $.- One vertex is at the origin (0,0,0).- The opposite vertex lies on the plane $ x + 3y + 6z = 36 $.The volume $ V $ of a rectangular box with sides $ x, y, $ and $ z $ is given by:\[ V = xyz \]We also need to satisfy the constraint imposed by the plane:\[ x + 3y + 6z = 36 \]To maximize the volume, you can use the method of Lagrange multipliers or substitute the constraint into the volume formula and find the critical points.Once you set up the problem using these techniques, you find the optimal values of $ x, y, $ and $ z $ that maximize the volume under the given constraint.For educators, presenting this problem involves explaining the concepts of optimization, constraints, and how to apply calculus techniques to solve real-world problems. The solution involves algebraic manipulation, taking partial derivatives, and setting up equations to solve for the variables.No graphs or diagrams are provided, but you might find it useful to draw a 3D coordinate system with a plane to illustrate the problem visually.

Let x, y, z, represent the respective coordinates of the vertices on the coordinate planes. Since…

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Calculus

largest rectangular

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.CR: Chapter 9 Review

Problem 85CR

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Question

### Volume of the Largest Rectangular Box in the First Octant

**Problem Statement:**

Determine the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, one vertex at the origin, and its opposite vertex in the plane $ x + 3y + 6z = 36 $.

**Explanation:**

To solve this problem, you need to use calculus, specifically optimization techniques. Here, the goal is to maximize the volume of the rectangular box under given constraints. The constraints are:

- The box is in the first octant, meaning $ x, y, z \geq 0 $.
- One vertex is at the origin (0,0,0).
- The opposite vertex lies on the plane $ x + 3y + 6z = 36 $.

The volume $ V $ of a rectangular box with sides $ x, y, $ and $ z $ is given by:
\[ V = xyz \]

We also need to satisfy the constraint imposed by the plane:
\[ x + 3y + 6z = 36 \]

To maximize the volume, you can use the method of Lagrange multipliers or substitute the constraint into the volume formula and find the critical points.

Once you set up the problem using these techniques, you find the optimal values of $ x, y, $ and $ z $ that maximize the volume under the given constraint.

For educators, presenting this problem involves explaining the concepts of optimization, constraints, and how to apply calculus techniques to solve real-world problems. The solution involves algebraic manipulation, taking partial derivatives, and setting up equations to solve for the variables.

No graphs or diagrams are provided, but you might find it useful to draw a 3D coordinate system with a plane to illustrate the problem visually.

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