Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x +

Question

Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 9.

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Anna 2 weeks 2021-10-01T21:04:26+00:00 1 Answer 0

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    2021-10-01T21:05:54+00:00

    Answer: The volume of largest rectangular box is 4.5 units.

    Step-by-step explanation:

    Since we have given that

    Volume = xyz

    with subject to x+2y+3z=9

    So, let z=\dfrac{9-x-2y}{3}

    So, Volume becomes,

    V=xyz\\\\V=xy(\dfrac{9-x-2y}{3})\\\\V=\dfrac{9xy-x^2y-2xy^2}{3}

    Partially derivative wrt x and y we get that

    9-2x-2y=0\implies 2x+2y=9\\\\and\\\\9-x-4y=0\implies x+4y=9

    By solving these two equations, we get that

    x=3,y=\dfrac{3}{2}

    So, z=\dfrac{9-x-2y}{3}=\dfrac{9-3-3}{3}=\dfrac{3}{3}=1

    So, Volume of largest rectangular box would be

    xyz=3\times \dfrac{3}{2}\times 1=\dfrac{9}{2}=4.5

    Hence, the volume of largest rectangular box is 4.5 units.

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