Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid x^2/81+y^2/25+z^2/49=1

This site discusses the problem, as well as a trick shortcut:

http://math.stackexchange.com/questions/129249/maximize-volume-of-box-in-ellipsoid

google will provide other solutions as well.

To find the volume of the largest rectangular box that can be inscribed in the given ellipsoid, we can use an optimization technique.

Step 1: Understand the Problem
An ellipsoid is a three-dimensional shape that resembles an elongated sphere. In this case, the equation x^2/81 + y^2/25 + z^2/49 = 1 represents an ellipsoid centered at the origin, with semi-axes lengths of 9, 5, and 7 along the x, y, and z-axes, respectively. The problem asks us to find the volume of the largest rectangular box that can be inscribed within this ellipsoid.

Step 2: Determine the Constraints
Since we are looking for a rectangular box with edges parallel to the axes, we can assume that the sides of the box are aligned with the coordinate axes. Let the length, width, and height of the rectangular box be 2a, 2b, and 2c, respectively. We need to find the maximum values of a, b, and c such that the box can fit inside the given ellipsoid.

Step 3: Set up the Problem
Since the length of the box along the x-axis is 2a, the width along the y-axis is 2b, and the height along the z-axis is 2c, we can represent the coordinates of the box's vertices as (±a, ±b, ±c).

The equation of the ellipsoid can be used to determine the constraints on a, b, and c. Substituting the box's vertex coordinates into the equation, we get:

(a/9)^2 + (b/5)^2 + (c/7)^2 = 1

Step 4: Simplify the Constraints
We can simplify the equation by removing the squares:

(a/9)^2 + (b/5)^2 + (c/7)^2 = 1

Divide both sides by the respective coefficients:

a^2/81 + b^2/25 + c^2/49 = 1

Multiplying both sides by their respective denominators:

49a^2 + 81b^2 + 25c^2 = 81 * 49

Simplifying further:

49a^2 + 81b^2 + 25c^2 = 3969

Step 5: Optimize the Volume
The volume of the rectangular box is given by V = 8abc. We need to maximize this volume with the given constraint.

To optimize the volume, we can use the method of Lagrange multipliers. Set up the following system of equations:

1) ∂(8abc)/∂a = λ * ∂(49a^2 + 81b^2 + 25c^2 - 3969)/∂a
2) ∂(8abc)/∂b = λ * ∂(49a^2 + 81b^2 + 25c^2 - 3969)/∂b
3) ∂(8abc)/∂c = λ * ∂(49a^2 + 81b^2 + 25c^2 - 3969)/∂c
4) 49a^2 + 81b^2 + 25c^2 = 3969

Simplify each equation by taking partial derivatives:

1) 8bc = λ * 98a
2) 8ac = λ * 162b
3) 8ab = λ * 50c
4) 49a^2 + 81b^2 + 25c^2 = 3969

Solve this system of equations to find the values of a, b, and c.

Step 6: Calculate the Volume
Once you have the values of a, b, and c, substitute them into the volume formula V = 8abc. Calculate the volume to get the final answer.