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Math Help - Surface area of cuboid

  1. #1
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    Surface area of cuboid

    4 days till exams and counting down - nerves shattered already!
    here goes:

    A closed cuboid with rectangular base of width xcm. Length of base is twice the width and the volumn is 1944cm^3. The surface area of the cuboid is S cm^2.
    a) Show that S = 4x^2 + 5832x^-1.
    b) Given that x can vary, find the value of x that makes the surface area a minimum.
    c) Find the minimum value of the surface area.

    Going round in circles with this one. Or, going round in cuboids. ha ha
    I kill me!

    thanks all
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  2. #2
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    Quote Originally Posted by lemonz View Post
    4 days till exams and counting down - nerves shattered already!
    here goes:

    A closed cuboid with rectangular base of width xcm. Length of base is twice the width and the volumn is 1944cm^3. The surface area of the cuboid is S cm^2.
    a) Show that S = 4x^2 + 5832x^-1.
    b) Given that x can vary, find the value of x that makes the surface area a minimum.
    c) Find the minimum value of the surface area.

    Going round in circles with this one. Or, going round in cuboids. ha ha
    I kill me!

    thanks all
    I presume that "cuboid", here, is what I would call a "rectangular solid".

    Such a figure with "width" x, "length" y and "height" z has volume xyz and surface area 2xy+ 2xz+ 2yz.

    For this particular figure you are told that "Length of base is twice the width" so y= 2x.

    Also, "the volumn is 1944cm^3" so xyz= x(2x)z= 2x^2z= 1944 so that z= 1944/2x^2.

    (a) Since S= surface area= 2xy+ 2xz+ 2yz, y= 2x and z= 1994/2x^2, S= 2x(2x)+ 2x(1944/2x^2)+ 2(2x)(1944/2x^2) = 4x^2+ 1944/x+ 3888/x= 4x^2+ 5832/x= 4x^2+ 5832/x.

    (b) One way to find the maximum value of a function is to find where its derivative is [/tex]0: S'= 8x- 5832/x^2= 0[/tex] so  8x^3= 5832 and x^3= 729 and, finally, x= ^3\sqrt{729}= 9 cm..


    (c) S(9)= 4(9)^2+ 5832/9= 324+ 648= 972 cm^2
    Last edited by HallsofIvy; January 6th 2009 at 03:13 AM.
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  3. #3
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    Thanx

    Wow!
    I have to say, that was a bit more than I expected. But, I worked my way through it and understand fully, now.
    Thank you so much.
    Regards
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