The box is constructed using a sheet that has dimensions 3m*5m. A square is cut from each corner and the sheet folded to create the box.

Let the side of the square that is cut be x.

The volume of the box is (3 - 2x)(5 - 2x)*x

V =...

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The box is constructed using a sheet that has dimensions 3m*5m. A square is cut from each corner and the sheet folded to create the box.

Let the side of the square that is cut be x.

The volume of the box is (3 - 2x)(5 - 2x)*x

V = x(15 - 10x - 6x + 4x^2)

=> 15x - 16x^2 + 4x^3

The maximum value of V can be determined by solving V' = 0

V' = 15 - 32x + 12x^2

15 - 32x + 12x^2 = 0

The quadratic equation 12x^2 - 32x + 15 = 0 has roots

x1 = `32/24 + sqrt(1024 - 720)/24`

=> `8/6 + sqrt 19/6`

x2 = `8/6 - sqrt 19/6`

The volume of the box for `8/6 + sqrt 19/6` is negative. For x = `4/3 - sqrt 19/6` the volume of the box is approximately 4.1044 m^3

**The maximum volume of the box is approximately 4.1044 m^3**