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21 June, 10:05

Arlan needs to create a box from a piece of cardboard. The dimensions of his cardboard are 10 inches by 8 inches. He must cut a square from each corner of the cardboard, in order to form a box. What size square should he cut from each corner, in order to create a box with the largest possible volume?

A. 0.5 inches

B. 1 inch

C. 1.5 inches

D. 2 inches

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  1. 21 June, 10:33
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    1) Dimensiones of the cardboard:

    length: 10 inches

    width: 8 inches

    2) dimensions of the squares cut

    length: x

    width: x

    3) dimensions of the box:

    length of the base = 10 - 2x

    width of the base = 8 - 2x

    height = x

    4) Volume of the box

    V = (10 - 2x) (8 - 2x) x = x [80 - 20x - 16x + 4x^2] = x [ 80 - 36x + 4x^2 ] =

    V = 80x - 36x^2 + 4x^3

    5) Maximum volume = > derivative of V, V' = 0

    V' = 80 - 72x + 12x^2 = 0

    6) Solve the equation

    Divide by 4 = > 3x^2 - 18x + 20 = 0

    Use the quadratic formula: x = 1.47 and x = 4.53 (this is not valid)

    So, the answer is the option C. 1.5 inches.
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