 My Math Forum Partial Derivative Application for 3D object

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 March 21st, 2017, 07:54 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 Partial Derivative Application for 3D object Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 2x^2 + 72y^2 + 18z^2 = 288. Answer: 256/sqrt(3) I would be much appreciated if anyone kindly point me to the right direction. Thanks! March 21st, 2017, 08:33 PM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,648 Thanks: 1476 well by symmetry your corner points will be $(\pm a, \pm b, \pm c)$ with volume $V=8abc$ you can reduce this a bit using the formula for the ellipsoid to get points at $\left(\pm a, \pm b, \pm \sqrt{\dfrac{288 - 2a^2 - 72b^2}{18}}\right)$ with volume $V=8ab \sqrt{\dfrac{288 - 2a^2 - 72b^2}{18}}$ As usual take the gradient of this expression, with respect to $a,~b$ and set it equal to the vector 0 and solve for $a,~b$ Solving this might be a bit of a mess but Mathematica returns a pretty digestible answer so it might not be that bad. The way to ensure that this extreme points is a maximum is a bit more complicated for a 3D than a 2D case. Do you know how to use the two dimensional 2nd derivative test to do this? Last edited by romsek; March 21st, 2017 at 09:13 PM. Tags application, derivative, object, partial Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zollen Calculus 1 March 20th, 2017 06:47 PM zollen Calculus 5 March 14th, 2017 05:21 AM hyperbola Calculus 3 May 3rd, 2015 04:41 AM sachinrajsharma Calculus 1 May 18th, 2013 08:36 PM EXPLORE Calculus 2 January 28th, 2010 03:03 AM

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