contestada

A rectangular 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

Respuesta :

Edufirst

Answer:

  • $ 163.54

Explanation:

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

  • Base area, B = base length × base width

       base width, w = x

       base length, l = 2x

        B = (2x) (x) = 2x²

  • Volume, V = base area × height = 10 m³

        height = h

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

2) Total area, A

  • Base, B = 2x²

  • Side 1, S₁

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

  • Side 2, S₂

        S₂ = S₁ = 5 / x

  • Side 3, S₃

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

  • Side 4, S₄

        S₄ = S₃ = 10 / x

3) Cost

  • Material for the base:

        $ 10 (2x²) = 20x²

  • Material for the sides

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

  • Total cost = 20x² + 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.

  • Derivative: (20x² + 180 / x)' = 40x - 180 / x² = 0

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

  • 40x - 180 / x² = 0

  • Assume x ≠ 0 and multiply by x² :  40x³ - 180 = 0

  • Add 180 to both sides: 40x³ =  180

  • Divide by 40: x³ = 4.5

  • Cubic root: x = 1.65

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

  • 20x² + 180 / x = 20 (1.65)² + 180 / (1.65) = 163.54

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

Otras preguntas