Answer to Question #274266 in Calculus for 1234

Question #274266

What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoid x2/9+y2/16+z2/36=1? Show that the answer you get gives you the largest volume.(DO NOT USE LAGRANGE MULTIPLIERS)


1
Expert's answer
2021-12-05T17:41:54-0500

Let P=(x,y,z) be a point on the ellipsoid with x,y,z>0. Take the eight different points with

Pi​(±xyz)

These points are the vertices of a parallelepiped with the side length

2x,2y and 2z

Then, the volume parallelepiped is:

"V=2x\u22c52y\u22c52z=8xyz"

"z=6\\sqrt{1-x^2\/9-y^2\/16}"


"V=48xy\\sqrt{1-x^2\/9-y^2\/16}"


"V_x=48y\\sqrt{1-x^2\/9-y^2\/16}-\\frac{16x^2y}{3\\sqrt{1-x^2\/9-y^2\/16}}=0"


"9(1-x^2\/9-y^2\/16)-x^2=0"


"V_y=48x\\sqrt{1-x^2\/9-y^2\/16}-\\frac{16y^2x}{3\\sqrt{1-x^2\/9-y^2\/16}}=0"


"9(1-x^2\/9-y^2\/16)-y^2=0"


"x^2=y^2"

"x=y"

"9-2x^2-9x^2\/16=0"

"41x^2=144"

"x=y=\\sqrt{41}\/12"


"V_{xx}=48y\\sqrt{1-x^2\/9-y^2\/16}-\\frac{16x^2y}{3\\sqrt{1-x^2\/9-y^2\/16}}="


"=-\\frac{32xy}{3\\sqrt{1-x^2\/9-y^2\/16}}"


"z=6\\sqrt{1-x^2\/9-x^2\/16}=\\sqrt{144-25x^2}\/2=\\sqrt{19711}\/24"


"V_{max}=\\frac{8\\cdot41\\sqrt{19711}}{144\\cdot24}=13.32"



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