\"Show that the rectangular box of maximum volume inscriptions in a sphere of radius $r$ is a cube\"

I addressed this using Lagrange. However I think this have the right to be fixed using partial derivatives. However, I\"m not completely sure about this. So ns would choose to know if this is possible.

You are watching: Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.

Sphere is given by $x^2 + y^2 + z^2 = r^2$.

For the rectangle-shaped box with facility at the origin,

$V = 8 xyz = 8xy \\sqrtr^2 - x^2 - y^2$

$\\frac\\partial V\\partial x = 8y \\sqrtr^2 - x^2 - y^2 - \\frac8x^2y\\sqrtr^2 - x^2 - y^2$

To find an essential points, we equate it to zero.

As $\\sqrtr^2 - x^2 - y^2 \\ne 0$ in a sphere, we obtain $8y (r^2 - x^2 - y^2) - 8x^2y = 0$

Again as $y \\ne 0$, $2x^2 + y^2 = r^2$.

Similarly indigenous the various other partial derivative (wrt $y$), we get $2y^2 + x^2 = r^2$

Solving them for $x, y$ and also then plugging right into equation of sphere to find $z$, us will notification they are all equal and also of size $\\frac2r\\sqrt3$.

share

point out

monitor

reply Nov 18 \"20 in ~ 21:54

carolannpeacock.com Lovercarolannpeacock.com Lover

34.9k33 yellow badges1111 silver- badges4040 bronze badges

$\\endgroup$

5

include a comment |

## her Answer

Thanks for contributing solution to carolannpeacock.comematics ridge Exchange!

Please be certain to*answer the question*. Provide details and share her research!

But *avoid* …

Use carolannpeacock.comJax to layout equations. Carolannpeacock.comJax reference.

To learn more, watch our tips on writing an excellent answers.

See more: Stripper Pick Up Walkthrough S, Stripper Pick Up

Draft saved

Draft discarded

### Sign up or log in

sign up utilizing Google

sign up using Facebook

sign up utilizing Email and also Password

submit

### Post together a guest

surname

email Required, however never shown

### Post as a guest

surname

Required, however never shown

short article Your answer Discard

By click “Post your Answer”, friend agree come our terms of service, privacy policy and cookie policy

## Not the prize you're spring for? Browse other questions tagged multivariable-calculus optimization or asking your own question.

Featured ~ above Meta

31 votes · comment · stats

Linked

0

Maximize volume of square inside unit sphere

0

Maximum volume that closed rectangular box that can be enrolled in unit sphere

connected

6

how to maximize the volume of a rectangle-shaped parallelepiped in one ellipsoid?

3

Optimization - discover the size of a box as functions of volume - minimal surface ar area

1

maximum Volume the a rectangle-shaped box in ellipsoid

0

Maxima and minima the parallelopiped

2

how to uncover the maximal volume of a rectangular box offered a addressed surface area?

2

find rectangular crate which has greatest volume

1

Finding smallest Radius that a sphere that can Inscribe a circular Cylinder

3

Cylinder of biggest Volume inscriptions in a Cube with its Axis as the diagonal

warm Network questions much more hot inquiries

concern feed

i ordered it to RSS

inquiry feed To subscribe to this RSS feed, copy and also paste this URL right into your RSS reader.

carolannpeacock.comematics

firm

stack Exchange Network

site architecture / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.9.16.40224

carolannpeacock.comematics ridge Exchange works finest with JavaScript enabled

your privacy

By clicking “Accept all cookies”, you agree stack Exchange have the right to store cookie on your machine and disclose details in accordance v our Cookie Policy.