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Math Help - story problem PLEASE HELP...

  1. #1
    gw16
    Guest

    story problem PLEASE HELP...

    Ok this chapter we are using vertex form to solve story problems.

    Here is the Prob

    A square, which is 5cm by 5cm, is cut from each corner of a rectangular piece of cardboard snd the sides are folded up to make a box. If the bottom of the box must have a perimiter of 50cm, what would be the length, width and height for the maximum volume?

    Thanks please help i need to see all the steps

    thanks

    Greg
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
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    A square, which is 5cm by 5cm, is cut from each corner of a rectangular piece of cardboard snd the sides are folded up to make a box. If the bottom of the box must have a perimiter of 50cm, what would be the length, width and height for the maximum volume
    Maximum volume?
    Okay.
    Box.
    V = L*w*h

    The cut squares at each corner of the rectangular cardboard are 5cm by 5cm each. So if the sides of the box are folded up, the h of the box is 5cm.
    Then the bottom of the box is L by w, in cms.

    "If the bottom of the box must have a perimiter of 50cm,..."
    L+w+L+w = 50
    2L +2W = 50
    L +w = 25 -------(i)

    Hence, V = L*w*5 = 5Lw -----(ii)
    So that we will have one variable only for V, say we eliminate L.
    From (i), L = 25 -w. Substitute that into (ii),
    V = 5(25 -w)w
    V = 125w -5w^2 ----(iii)
    Differentiate both sides of (iii) with respect to w,
    dV/dw = 125 -10w
    Set dV/dw to zero,
    0 = 125 -10w
    10w = 125
    w = 125/10 = 12.5 cm.
    Hence,
    L = 25 -w = 25 -12.5 = 12.5 cm also.

    Therefore, for maximum volume, length is 12.5cm, width is 12.5cm, and height is 5cm. ------answer.
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