Max Vol Q: Find the Volume of Rectangular Box Inscribed in Ellipsoid

In summary, the question asks to find the volume of the largest rectangular box that can be inscribed in the ellipsoid 9x^2+36y^2 + 4z^2 = 36. The method to solve this problem is to maximize the function V = 8xyz while satisfying the condition 9x^2 + 35y^2 + 4z^2 = 36. This can be done using the Lagrange multiplier method, which involves taking partial derivatives of both the function and the condition and setting them equal to each other. The resulting equations can then be solved to find the values of x, y, and z that maximize the volume.
  • #1
pezzang
29
0
"Hi, I have a question on max vol. q. Its invloved with multivariable calculus.

Q) Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 9x^2+36y^2 + 4z^2 = 36.

What i did was i found the three x,y and z-intersection points.
(2,0,0), (0,1,0), and (0,0,3)

Then, I just assumed the following equation:

(2x)/4 + y +(3z)/9 = 1. <- I substituted value of x,y and z for the intersection value.
And if i simplify it, i get: z = 3 - 3x/4 - 3y

To find volume,

V = xyz

so,
V = xy(3 - 3x/4 - 3y)
and fsubx = 3y-3xy-3y^2 = 0
fsuby = 3x - (3x^2)/2 -6xy = 0.

If i do the calculation

i get: 6y - 3x -3y^2 +3(x^2)/2 = 0.

and x = 2/3 and y = 1/3.

And if i sub 2/3 and 1/3 for x and y in the original equation, i get
V = 2/9.

IS THIS THE RIGHT WAY TO DO IT? I ASSUMED THE BEGINNING PART OF THE PROBLEM SOLVING SO I MIGHT BE COMPLETELY WRONG. ANY OF YOU MATH EXPERT, PLEASE HELP ME OUT! THANK YOU SO MUCH AND HAVE A NICE DAY!

(PLEASE PROVIDE ME THE COMPLETE ANSWER AND EXPLANATION ON THE QUETSION.)

AGAIN THANK YOU SO MUCH!
 
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  • #2
What you are calling the "intersection points" are the vertices of the ellipsoid. There is no point on the ellipsoid having those x y z values as coordinates.


Let (x,y,z) be the vertex of the rectangular box in the first octant. Then the box has dimensions 2x by 2y by 2z and its volume is
V= 8xyz. That's the function you want to maximize. The vertices all lie on the ellipsoid so we must have 9x2+ 35y2+ 4z2= 36.

That is, the problem is to maximize 8xyz subject to the condition
9x2+ 35y2+ 4z2= 36.

One way of doing this is to solve the subsidiary equation for one of the variables and put that into the "object" function V= 8xyz to reduce to two variables. Then take the partial derivatives with respect to those two variables and set them equal to 0.

A better way (if you haven't already learned it, you should soon) is to use the "Lagrange multiplier" method. Since we must remain in a given set (the subsidiary equation), we do not require that the derivative (gradient) be 0 but that it be perpendicular to the set.

Specifically, here, we would take partial derivatives of V= 8xyz:
Vx= 8yz, Vy= 8xz, Vz= 8xy. The partial derivatives of the subsidiary equation (and so the perpendicular to it) are 18x, 70y, and 8z. The x, y, z that maximize the volume of this rectangle satisfy 8yz= 18 lambda x, 8xz= 70lambda y, 8xy= 8lambda z for some number lambda. Those three equations, together with 9x2+ 35y2+ 4z2= 36 can be solved for x, y, z.
 
  • #3
"

Hi there,

Yes, your approach to finding the volume of the largest rectangular box inscribed in the given ellipsoid is correct. Let's break down the steps and provide a complete answer and explanation for the question.

First, we need to understand what is meant by an ellipsoid. An ellipsoid is a three-dimensional shape that is formed by rotating an ellipse about one of its axes. It can be described by the equation (x^2/a^2) + (y^2/b^2) + (z^2/c^2) = 1, where a, b, and c are the lengths of the semi-axes along the x, y, and z directions respectively. In this case, we have the equation 9x^2 + 36y^2 + 4z^2 = 36, which can be rewritten as (x^2/4) + (y^2/1) + (z^2/9) = 1. This tells us that a = 2, b = 1, and c = 3.

Next, we need to find the three intersection points of the ellipsoid with the x, y, and z axes. You correctly found these points to be (2,0,0), (0,1,0), and (0,0,3). These points will serve as the lengths of the rectangular box along the x, y, and z directions respectively.

Now, we need to find the equation for the plane that contains these three points. To do this, we can use the fact that a plane can be described by the equation Ax + By + Cz = D, where A, B, and C are the coefficients of x, y, and z respectively, and D is a constant. We can plug in the values of our three points to get a system of equations:

2A + 0B + 0C = D
0A + 1B + 0C = D
0A + 0B + 3C = D

Solving this system, we get A = 2/3, B = 1/3, C = 1, and D = 2/3. Therefore, the equation of the plane containing the three points is (2/3)x + (1/3)y + z = 2/3.

Now, we can use this equation to find the length of
 

Related to Max Vol Q: Find the Volume of Rectangular Box Inscribed in Ellipsoid

1. How do you determine the volume of a rectangular box inscribed in an ellipsoid?

The volume of a rectangular box inscribed in an ellipsoid can be found by multiplying the length, width, and height of the box. The length, width, and height can be determined by finding the points where the box intersects with the ellipsoid's major, minor, and intermediate axes.

2. What is an ellipsoid?

An ellipsoid is a three-dimensional figure that resembles a flattened sphere. It is defined by three axes - major, minor, and intermediate - that intersect at the center of the figure. A real-life example of an ellipsoid is an egg.

3. Is it possible to find the volume of a rectangular box inscribed in any ellipsoid?

Yes, it is possible to find the volume of a rectangular box inscribed in any ellipsoid as long as the ellipsoid's axes are known. The formula for the volume of a rectangular box inscribed in an ellipsoid is V = 4/3 * pi * a * b * c, where a, b, and c are the lengths of the major, minor, and intermediate axes.

4. Can you explain the concept of inscribed boxes in an ellipsoid?

Inscribed boxes in an ellipsoid refer to rectangular boxes that are placed inside the ellipsoid in a way that each of their eight vertices touch the ellipsoid's surface. These boxes are unique for each ellipsoid and can be used to find the volume of the ellipsoid.

5. Are there any real-life applications of finding the volume of a rectangular box inscribed in an ellipsoid?

Yes, there are many real-life applications of finding the volume of a rectangular box inscribed in an ellipsoid. For example, this concept is used in engineering and architecture to determine the volume of objects such as bridges, tunnels, and buildings that have an ellipsoidal shape. It is also used in geodesy to calculate the volume of the Earth's geoid, which is an ellipsoid-shaped representation of the Earth's surface.

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