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.)

Refer to the diagram shown below.

The volume of the container is 10 m³, therefore
x*2x*h = 10
2x²h = 10
h = 5/x²        (1)

The base area is 2x² m².
The cost is $10 per m², therefore the cost of the base is
(2x²)*($10) = 20x²

The area of the sides is
2hx + 2(2xh) = 6hx = 6x*(5/x²) = 30/x m²
The cost is $6 per m², therefore the cost of the sides is
(30/x)*($6) = 180/x

The total cost is
C = 20x² + 180/x

The minimum cost is determined by C' = 0.
That is,
40x - 180/x² = 0
x³ = 180/40 = 4.5
x = 1.651
The second derivative of C is
C'' = 40 + 360/x³
C''(1.651) = 120 >0, so x = 1.651 m yields the minimum cost.

The total cost is
C = 20(1.651)² + 180/1.651 = $163.54

$163.54