Answer:
Dimensions:
l =3,15 m
w=1,15 m
x= 0,42 m (height)
V(max) = 1,52 m³
Step-by-step explanation:
The cardboard is L = 4 m and W = 2 m
Let call x the length of the square to cut in each corner, then, volume of open box is:
For the side L is L - 2*x l = 4 - 2*x
For the side W is W - 2*x w= 2 - 2*x
The height is x
Volume of the open box, as function of x is:
V(x) = ( 4 -2x) * ( 2 - 2x) *x ⇒ V(x) = ( 8 - 8x -4x + 4x²) *x
V(x) = ( 8 -12x + 4x² )*x V(x) = 8x - 12x² + 4x³
V(x) = 8x - 12x² + 4x³
Taking derivatives on both sides of the equation
V´(x) = 8 - 24x + 12x²
V´(x) = 0 8 - 24x + 12x² = 0 reordering 12x² - 24x + 8 = 0
or 3x² - 6x + 2 = 0
A second degree equation. Solving for x
x₁,₂ = ( 6 ± √36 - 24 ) /6
x₁,₂ = ( 6 ± 3.46) /6
x₁ = 6 + 3,46 /6 x₁ = 1.58 we dismiss such solution because 1,58 * 2 = 3,15 and is bigger than 2 one of the side of the cardboard
x₂ =( 6 - 3,46 ) / 6
x₂ = 0,42 m
Dimensions of the open box
l = 4 - 2*x l = 4 - 0,85 l = 3,15 m
w = 2 -2*x w = 2 - 0,85 w = 1,15 m
x = 0,42 m
V(max) =3,15*1,15*0,42
V(max) = 1,52 m³
