Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

. A rectangular box, whose edges are parallel to the coordinate axes, is inscribed in the ellipsoid \(96 x^{2}+4 y^{2}+4 z^{2}=36\). What is the greatest possible volume for such a box?

Step-by-Step Solution

Answer

The greatest possible volume is \(\frac{9\sqrt{6}}{8}\).

Step 1: Understand the Problem

We need to find the maximum volume of a rectangular box inscribed in the ellipsoid given by the equation \(96x^2 + 4y^2 + 4z^2 = 36\). The box's edges are aligned with the coordinate axes, meaning it is symmetrically placed within the ellipsoid.

Step 2: Simplify the Ellipsoid Equation

First, we rewrite the equation of the ellipsoid in a more convenient form by dividing through by 36: \( \frac{x^2}{3/8} + \frac{y^2}{9} + \frac{z^2}{9} = 1 \). This represents an ellipsoid centered at the origin with semi-axis lengths \(a = \sqrt{3/8}, b = 3, c = 3\).

Step 3: Consider the Box Dimensions

An inscribed rectangular box in the ellipsoid has corners at \((\pm x_0, \pm y_0, \pm z_0)\), and the ellipsoid equation at these points is \( \frac{x_0^2}{3/8} + \frac{y_0^2}{9} + \frac{z_0^2}{9} = 1 \). The volume of the box is \(V = 8x_0y_0z_0\), as each dimension is doubled.

Step 4: Express Volume in Terms of Ellipsoid Constraint

Given the constraint \( \frac{x_0^2}{3/8} + \frac{y_0^2}{9} + \frac{z_0^2}{9} = 1 \), we aim to maximize \( V = 8x_0y_0z_0 \) under this condition.

Step 5: Use Lagrange Multipliers to Maximize Volume

To apply the method of Lagrange multipliers, let \(f(x_0, y_0, z_0) = x_0 y_0 z_0\) and \(g(x_0, y_0, z_0) = \frac{x_0^2}{3/8} + \frac{y_0^2}{9} + \frac{z_0^2}{9} - 1 = 0\). The gradients need to satisfy: \(abla f = \lambda abla g\).

Step 6: Compute the Gradients

Calculate the gradients: \( abla f = (y_0 z_0, x_0 z_0, x_0 y_0) \) and \( abla g = \left( \frac{16x_0}{3}, \frac{2y_0}{9}, \frac{2z_0}{9} \right) \). Equate components: \( \frac{16x_0 \lambda}{3} = y_0 z_0 \), \( \frac{2y_0 \lambda}{9} = x_0 z_0 \), \( \frac{2z_0 \lambda}{9} = x_0 y_0 \).

Step 7: Solve Lagrange Equations Simultaneously

Simultaneously solve the system of equations from gradients to find \(x_0, y_0, z_0\). Set ratios for symmetry: \(x_0 = ky_0\) and \(x_0 = kz_0\). Then, substitute back into the ellipsoid constraint for one variable and solve.

Step 8: Find Maximum Volume

By solving the system of equations, determine \(x_0 = \frac{1}{2} \sqrt{3/8} \), \(y_0 = \frac{3}{2} \), and \(z_0 = \frac{3}{2}\). Calculate: \(V = 8 \cdot \frac{1}{2}\sqrt{3/8} \cdot \frac{3}{2} \cdot \frac{3}{2}\).

Step 9: Calculate and Present the Final Volume

Substitute to find the greatest volume: \(V = 8 \cdot \frac{1}{2} \cdot \sqrt{3/8} \cdot \frac{9}{4} = \frac{9 \sqrt{6}}{8}\).

Key Concepts

Ellipsoid

An ellipsoid is a three-dimensional shape that resembles a stretched or flattened sphere. In mathematics, it is defined by an equation similar to the one used for ellipses. In this exercise, the ellipsoid is represented by the equation \[ 96x^2 + 4y^2 + 4z^2 = 36 \]. This equation can be simplified by dividing the whole expression by 36, resulting in\[ \frac{x^2}{3/8} + \frac{y^2}{9} + \frac{z^2}{9} = 1. \] This simplification helps to see how the ellipsoid stretches along different axes. The squared terms in the equation correspond to the semi-axis lengths:

• \( a = \sqrt{3/8} \) along the x-axis
• \( b = 3 \) along the y-axis
• \( c = 3 \) along the z-axis

These values tell us how far the surface of the ellipsoid extends along each axis from the center of the ellipsoid, which is at the origin. Understanding the dimensions of the ellipsoid is crucial for determining how a rectangular box can fit inside it.

Lagrange Multipliers

Lagrange multipliers are a powerful mathematical tool used when optimizing a function subject to a constraint. Here, they help us find the maximum volume of a box inscribed in an ellipsoid. The method introduces an auxiliary variable, usually denoted by \( \lambda \), to merge both the objective function and the constraint into a system that can be solved.
The primary goal is to maximize the volume \( V = 8x_0y_0z_0 \). Meanwhile, the constraint is represented by the ellipsoid equation:\[ \frac{x_0^2}{3/8} + \frac{y_0^2}{9} + \frac{z_0^2}{9} = 1. \]Applying the Lagrange multipliers method involves:

• Defining \( f(x_0, y_0, z_0) = x_0 y_0 z_0 \) as the function to be maximized.
• Using \( g(x_0, y_0, z_0) = \frac{x_0^2}{3/8} + \frac{y_0^2}{9} + \frac{z_0^2}{9} - 1 = 0 \) as the constraint.
• Finding the gradients of both functions: \( abla f = (y_0 z_0, x_0 z_0, x_0 y_0) \) and \( abla g = \left( \frac{16x_0}{3}, \frac{2y_0}{9}, \frac{2z_0}{9} \right) \).

Equating \( abla f = \lambda abla g \) allows us to create a system of equations to solve for \( x_0, y_0, z_0 \). This method finds the critical points that lead to the maximum possible volume under the given constraint.

Rectangular Box Volume Maximization

Volume maximization of a rectangular box involves finding the largest possible volume such a box can have when inscribed within another shape, in this case, an ellipsoid. For a rectangular box aligned with the coordinate axes, its corners are located at \( (\pm x_0, \pm y_0, \pm z_0) \).
The volume \( V \) of the box is given by the formula:\[ V = 8x_0y_0z_0. \]The problem becomes one of identifying the dimensions \( x_0, y_0, z_0 \) that will both fit within the ellipsoid and make \( V \) as large as possible.

• Use symmetry to simplify calculations: set \( x_0 = ky_0 \) and \( x_0 = kz_0 \), leveraging the symmetry observed along the axes.
• Substitute these into the constraint to express \( y_0 \) (or \( z_0 \)) in terms of the others, reducing variables.
• Solve the resulting equations to find the specific values of \( x_0, y_0, z_0 \) that maximize \( V \).

The calculated maximum volume considers both the shape of the box and the constraint imposed by the ellipsoid, resulting in a volume, in this case, \( V = \frac{9 \sqrt{6}}{8} \), that is optimal within the prescribed boundary conditions.