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28 May, 22:37

According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 98 in. Find the dimensions of a carton with square base that is not oversized and has maximum volume.

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  1. 29 May, 01:55
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    V (max) = 8712.07 in³

    Dimensions:

    x (side of the square base) = 16.33 in

    girth = 65.32 in

    height = 32.67 in

    Step-by-step explanation:

    Let

    x = side of the square base

    h = the height of the postal

    Then according to problem statement we have:

    girth = 4*x (perimeter of the base)

    and

    4 * x + h = 98 (at the most) so h = 98 - 4x (1)

    Then

    V = x²*h

    V = x² * (98 - 4x)

    V (x) = 98*x² - 4x³

    Taking dervatives (both menbers of the equation we have:

    V' (x) = 196 x - 12 x² ⇒ V' (x) = 0

    196x - 12x² = 0 first root of the equation x = 0

    Then 196 - 12x = 0 12x = 196 x = 196/12

    x = 16,33 in ⇒ girth = 4 * (16.33) ⇒ girth = 65.32 in

    and from equation (1)

    y = 98 - 4x ⇒ y = 98 - 4 (16,33)

    y = 32.67 in

    and maximun volume of a carton V is

    V (max) = (16,33) ² * 32,67

    V (max) = 8712.07 in³
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