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22 May, 15:05

If 1200 cm^2 of material is available to make a box with a square base and an open top find the largest possible volume of the box

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  1. 22 May, 16:44
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    Surface area of box=1200 cm²

    Volume of box=s²h

    s = side of square base

    h = height of box

    S. A. = s² + 4sh

    S. A. = surface area or 1200 cm², s²

    = the square base, and 4sh = the four 'walls' of the box.

    1200 = s² + 4sh

    1200 - s² = 4sh

    (1200 - s²) / (4s) = h

    v (s) = s² ((1200 - s²) / (4s))

    v (s) = s (1200 - s²) / 4.

    v (s) = 300s - (1/4) s^3

    by derivating

    v' (s) = 300 - (3/4) s²

    0 = 300 - (3/4) s²

    -300 = (-3/4) s²

    400 = s²

    s = - 20 and 20.

    again derivating

    v" (s) = - (3/2) s

    v" (-20) = - (3/2) (-20)

    v" (-20) = 30

    v" (20) = - (3/2) (20)

    v" (20) = - 30

    v (s) = 300s - (1/4) s^3

    v (s) = 300 (20) - (1/4) (20) ^3

    v (s) = 6000 - (1/4) (8000)

    v = 6000 - 2000

    v=4000
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