Answer:
side of the square x = 2.43 in
Area of the square x² = 5.91 in²
Step-by-step explanation:
The piece of cardboard is
20 in * 12 in (rectangle) we are going to cut four squares each in a corner of that piece then
V(b) = ( 12 - 2x )* ( 20 - 2x) *x
V(b) = ( 240 - 24x - 40x + 4x² ) *x V(b) = ( 240 - 64x + 4x²)*x
V(b) = 240x - 64x² + 4x³
Taking derivatives both sides of equation
V´(b) = 240 - 128x + 12x²
Then V´(b) = 0 240 - 128x + 12x² = 0
A second degree equation solving we have
12x² -128x + 240 = 0 ⇒ 3x² - 32x + 60
x₁,₂ = [ 32 ± √(32)² - 720] / 6 ⇒ x₁,₂ = [ 32 ± 17.44]/6
x₁ = 8.24 in we dismiss this value since 2 times this value is bigger than one side which is not possible
x₂ = 2.43 in then the square is (2.43)² = 5.91 in²
