Answer to Question #274322 in Calculus for Sidrat

Question #274322

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
2021-12-14T13:48:39-0500

let (x,y,z) be the coordinates of the corner of the box that is inscribed in an ellipsoid. Thus "x,y,z \\geq0" and equation of ellipsoid is

"\\frac{x^2}{9}+\\frac{y^2}{16}+\\frac{z^2}{36}=1"


"\\frac{z^2}{36}=1-\\frac{x^2}{9}-\\frac{y^2}{16}"

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


The volume of the box is represented as

"V=(2x)(2y)(2z)=48xy\\sqrt{1-\\frac{x^2}{9}-\\frac{y^2}{16}}"

For "x\\geq0,y\\geq0, and \\ (\\frac{x^2}{a^2})+(\\frac{y^2}{b^2})\\leq1"


For analysis it is easier to deal with "V^2\\ than\\ V" as it becomes more complex to find the partial derivative in terms of x and y

"V^2=2304[x^2y^2-\\frac{x^4y^2}{9}-\\frac{x^2y^4}{16})"


Since "V=0 \\ if \\ x=0 \\ or \\ y=0 \\ or\\ (\\frac{x^2}{a^2})+(\\frac{y^2}{b^2})=1,"

the maximum value of V2 and hence of V will occur at critical point of V2 to find the critical point we we have to find the partial derivative in terms of x and y


"\\frac{\\delta V^2}{\\delta x}=2304[2xy^2-\\frac{4x^3y^2}{9}-\\frac{2x^2y^4}{16})"


"=4608xy^2(1-\\frac{2x^2}{9}-\\frac{y^2}{16})........(1)"


similarly


"\\frac{\\delta V^2}{\\delta y}=4608xy^2(1-\\frac{x^2}{9}-\\frac{2y^2}{16})........(2)"


Now we will equate the above partial derivatives to zero so as to find critical points


"(1-\\frac{2x^2}{9}-\\frac{y^2}{16})=0\\\\\\frac{2x^2}{9}+\\frac{y^2}{16}=1........(3)"


"(1-\\frac{x^2}{9}-\\frac{2y^2}{16})=0\\\\\\frac{x^2}{9}+\\frac{2y^2}{16}=1........(4)"


Now by equating (3) and (4) we get


"\\frac{2x^2}{9}+\\frac{y^2}{16}=1=\\frac{x^2}{9}+\\frac{2y^2}{16}"


which

"\\frac{x^2}{9}=\\frac{1}{3}\\implies x=\\frac{9}{\\sqrt{3}}"


"\\frac{y^2}{16}=\\frac{1}{3}\\implies y=\\frac{16}{\\sqrt{3}}"


The volume of the largest box that can be inscribed inside the ellipsoid is


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


"=48(\\frac{9}{\\sqrt{3}})(\\frac{16}{\\sqrt{3}})\\sqrt{1-\\frac{1}{3}-\\frac{1}{3}}"


"=1330.215"

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