Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 11 in. by 7 in. by cutting cong

Question

Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 11 in. by 7 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume.

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Remi 2 months 2021-09-17T15:55:50+00:00 1 Answer 0 views 0

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    2021-09-17T15:56:53+00:00

    Answer:

    The dimension of the open rectangular box is 8.216\times 4.216\times 1.392.

    The volume of the box is 8.217 cubic inches.

    Step-by-step explanation:

    Given : The open rectangular box of maximum volume that can be made from a sheet of cardboard 11 in. by 7 in. by cutting congruent squares from the corners and folding up the sides.

    To find : The dimensions and the volume of the open rectangular box ?

    Solution :

    Let the height be ‘x’.

    The length of the box is ’11-2x’.

    The breadth of the box is ‘7-2x’.

    The volume of the box is V=l\times b\times h

    V=(11-2x)\times (7-2x)\times x

    V=4x^3-36x^2+77x

    Derivate w.r.t x,

    V'(x)=4(3x^2)-2(36x)+77

    V'(x)=12x^2-72x+77

    The critical point when V'(x)=0

    12x^2-72x+77=0

    Solve by quadratic formula,

    x=\frac{18+\sqrt{93}}{6},\frac{18-\sqrt{93}}{6}

    x=4.607,1.392

    Derivate again w.r.t x,

    V''(x)=24x-72

    Now, V''(4.607)=24(4.607)-72=38.568>0 (+ve)

    V''(1.392)=24(1.392)-72=-38.592<0 (-ve)

    So, there is maximum at x=1.392.

    The length of the box is l=11-2x

    l=11-2(1.392)=8.216

    The breadth of the box is b=7-2x

    b=7-2(1.392)=4.216

    The height of the box is h=1.392.

    The dimension of the open rectangular box is 8.216\times 4.216\times 1.392.

    The volume of the box is V=l\times b\times h

    V=8.216\times 4.216\times 1.392

    V=48.217\ in.^3

    The volume of the box is 8.217 cubic inches.

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