Answer:
x = 0.67 cm
Step-by-step explanation:
Let call " x " the length of the side of the square to cut from each corner
then the sides of the future box would be
L = 8 - 2x and D = 3 - 2x
The volume of the box is:
V = L*D*x
And such volume as function of x is
V(x) = ( 8 - 2x ) * ( 3 - 2x ) * x ⇒ V(x) = ( 24 - 16x - 6x + 4x²) * x
V(x) = 4x³ - 22x² + 24x
Taking derivatives on both sides of the equation we get:
V´(x) = 12x² - 44x + 24
Then V´(x) = 0 ⇒ 12x² - 44x + 24 = 0 ⇒ 3x² - 11x + 6 = 0
We got a second degree equation solving for x
x₁,₂ = [11 ± √ 121 - 72 ] / 6
x₁ = ( 11 + 7 ) / 6 x₁ = 3 we dismiss this solution since according to problem statement one side would become negative
Then
x₂ = ( 11 - 7 ) / 6 ⇒ x₂ = 4/6 ⇒ x₂ = 0.67 cm
As the second drivative is smaller than 0 then there is a maximun in that point
V´´(x) = 12x - 44 < 0
Sides of the box
L = 8 - 2x ⇒ L = 8 - 2*(0.67) ⇒ L = 8 - 1.34 ⇒ L = 6.66 cm
D = 3 - 2x ⇒ D = 3 - 2* (0.67) ⇒ D = 3 - 1.34 ⇒ D = 1.66 cm
Heigh = x = 0.67 cm
V(max) = 6.66*1.66*0.67
V(max) = 7.41 cm³
or
, where n is any natural number.