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15 July, 23:26

A rectangular storage container with an open top is to have avolume

20m3. The length of its base is twice the width. The

material for the base costs $5 per square meter. The materialfor

the sides costs $9 per square meter. Find the minimum cost tobuild

such container.

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Answers (1)
  1. 16 July, 00:32
    0
    Dimensions of the container:

    x = 3 m

    y = 6 m

    h = 1.1 m

    C (min) = 270 $

    Step-by-step explanation:

    Volume of storage container

    V = 20 m³

    Let "y" be the length and "x" the width then y = 2*x

    V = x*y*h ⇒ V = 2*x²*h ⇒ 20 = 2*x²+h ⇒ h = 10 / x²

    Costs:

    Total cost = cost of base (5*2*x²) + cost of side with base x (2*9*x*h) +

    cost of side witn base y = 2x (2*9*2x*h)

    C (t) = 10*x² + 18*x*h + 36*x*h

    C (x) = 10x² + 54*x*10/x² ⇒ C (x) 10*x² + 540 / x

    Taking derivatives on both sides of the equation we get:

    C' (x) = 20*x - 540/x²

    C' (x) = 0 ⇒ 20*x - 540/x² = 0 ⇒ 2x - 54/x² = 0

    2x³ - 54 = 0

    x³ = 27 x = 3 m

    Then y = 2*x ⇒ y = 2*3 y = 6 and h = 10 / x² h = 1.1 m

    And the minimum cost is

    C (min) = 10*x² + 540/x ⇒ C (min) = 90 + 180

    C (min) = 270 $
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