An open box is to be made from a rectangular piece of cardboard which is 12 inches by 20 inches by cutting equal squares from the corners and turning up the sides. Find the size of the square which gives the box of largest volume.

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