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An open cardboard box is to be made by cutting squares of side x cm from each corner of a card of side 60cm and folding the resulting "flaps" up to form the box. Find the value of x that gives the box a maximum capacity.

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1 answer
  1. dimension of the box is
    60-2x by 60-2x by x , where x < 30

    Vol = x(60-2x)^2
    = 3600x - 240x^2 + 4x^3
    d(Vol)/dx = 3600 - 480x + 12x^2
    = 0 for a max/min of Vol

    divide each term by 12
    x^2 - 40x + 300 = 0
    (x-10)(x-40) = 0
    x = 10 or x = 40, but x<30

    x = 10

    cut out flaps of 10 cm by 10 cm

    check:
    if x = 10 , Vol = 10(40)^2 = 16000
    if x = 10.1 , Vol = 10.1(39.8)^2 = 15998.8 which is < 16000
    8
    if x = 9.9 , Vol = 9.9(40.2)^2 = 15998.8 which is < 16000
    My answer is very plausible

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