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13 January, 11:17

A closed box with a square base is to have a volume of 13 comma 500 cm cubed. The material for the top and bottom of the box costs $10.00 per square centimeter, while the material for the sides costs $2.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total cost?

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  1. 13 January, 14:12
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    x = 1,5 cm

    h = 6 cm

    C (min) = 135 $

    Step-by-step explanation:

    Volume of the box is:

    V (b) = 13,5 cm³

    Aea of the top is equal to area of the base,

    Let call " x " side of the base then as it is square area is A₁ = x²

    Sides areas are 4 each one equal to x * h (where h is the high of the box)

    And volume of the box is 13,5 cm³ = x²*h

    Then h = 13,5/x²

    Side area is : A₂ = x * 13,5/x²

    A (b) = A₁ + A₂

    Total area of the box as functon of x is:

    A (x) = 2*x² + 4 * 13,5 / x

    And finally cost of the box is

    C (x) = 10*2*x² + 2.50*4*13.5/x

    C (x) = 20*x² + 135/x

    Taking derivatives on both sides of the equation:

    C' (x) = 40*x - 135*/x²

    C' (x) = 0 ⇒ 40*x - 135*/x² = 0 ⇒ 40*x³ = 135

    x³ = 3.375

    x = 1,5 cm

    And h = 13,5/x² ⇒ h = 13,5 / (1,5) ²

    h = 6 cm

    C (min) = 20*x² + 135/x

    C (min) = 45 + 90

    C (min) = 135 $
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