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