Arectangular storage container with an open top is to have a volume of 10 m3. the length of this base is twice the width. material for the base costs $10 per square meter. material for the sides costs $6 per square meter. find the cost of materials for the cheapest such container. (round your answer to the nearest cent. g

Explanation:

1) Write the model for the volume and base area of the rectangular sotorage container:

       base width, w = x

       base length, l = 2x

        B = (2x) (x) = 2x²

        height = h

        V = 2x² h = 10 ⇒ h = 10 / (2x²)

2) Total area, A

        S₁ = (x) . (h) = (x) . 10 / (2x²) = 10 / (2x) = 5 / x

        S₂ = S₁ = 5 / x

        S₃ = (2x) . (h) = (2x) . 10 / (2x²) = 10 / x

        S₄ = S₃ = 10 / x

3) Cost

        $ 10 (2x²) = 20x²

$6 (S₁ + S₂ + S₃ + S₄) = 6 (5/x + 5/x + 10/x + 10/x ) = 6 ( 30/x) = 180/x

4) Cheapest container

Minimum cost ⇒ find the minimum of the function 20x² + 180 / x, which formally is done by derivating the function and making the derivative equal to zero.

Solve to find the value of x that makes the first derivative equal to zero:

Replace x = 1.65 in the equations of costs to find the minimum cost:

That is the final answer, already rounded to the nearest cent: $163.54

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