Pre-calculus

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You are given a 12"x18" piece of construction paper. You are to cut a square out of each corner in order to create a 3-dimensional open-top box. What size squares would you need to cut in order to maximize the volume of the box?

  • Pre-calculus -

    let the side of the square to be cut out be x inches

    resulting box is (12-2x) by (18-2x) by x

    V = x(12-2x)(18-2x)= x(216 - 60x + 4x^2)
    = 4x^3 - 60x^2 + 216x

    dV/dx = 12x^2 - 120x + 216
    = 0 for a max of V
    x^2 - 10x + 18=0
    solve for x

  • Pre-calculus -

    length = (18 - 2x)
    width = (12 - 2x)

    V = x (18-2x)(12-2x)
    = x(4)(9-x)(6-x)
    = 4 x (54-15x+x^2)
    = 4 (54 x -15x^2 + x^3)
    dV/dx = 0 for max

    0 = 54 - 30 x + 3 x^2
    0 = 18 - 10 x + x^2
    x = [ 10 +/- sqrt (100-72) ]/2
    [ 10 +/- 2 sqrt 7 ]/2
    = 5 +/- sqrt 7
    5+sqrt 7 is no good, more than half the width
    so
    5-sqrt 7 = 2.35

  • Pre-calculus -

    Looks like calculus not pre-calculus to me by the way. I do not see how to do it without taking the derivative.

  • Pre-calculus -

    what does sqrt mean?

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