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a rectangular container with a square base and open top must have volume of 16 \text{} cubic meters. if material for the base costs $8 per square meter, and material for the sides costs $2 \text{} per square meter, find the dimensions of the container so that the cost of material to make it will be a minimum. as your answer, please input the value of the minimal cost.



Answer :

The minimal cost of the box is $96.

Given data;

The volume of the box (v) = x²y

Given that, volume = 16 m³

So, 16 = x²y

y = 16/x² → (I)

Material for the base costs $8 per square meter, and material for the sides costs $2 per square meter.

Total cost (C) = 8 (x²) + 2*(xy)(4)

                      = 8x² + 8xy

Using (I),

C(x) = 8x² + 8x(16/x²)

C (^) = 8x² + 128/x

Now find critical points from

∴ [tex]\frac{d}{dx} = [8x^2+\frac{128}{x} ]=0[/tex]

  x³ = 8

  x = 2

So, the minimal cost will be given by x= 2,

Now, y = 16/x²

            = 16/4

            = 4

Dimension of the box that minimizes the cost is;

height = 4m, width=2m and breadth = 12m

Now,

Minimum cost = 8 (2)² + 128/2 = 32 + 64

                        = 96

Hence, The minimal cost of the box is $96.

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