A box is to constructed from a sheet of cardboard that measure 3m x 5m. A square from each corner of the sheet will be cut then the sides will be folded to create the box. What is the greatest volume this box.
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