A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each corner and bending up the sides. What size square should be cut from each corner for the box to have the maximum volume?

x = 1.64 in the size of the side of the square

Step-by-step explanation:

Let call x side of the square to be cut from cornes, then:

First side of rectangular base

L = 14 - 2*x

And the other side

d = 8 - 2*x

Then Volume of the box

V (b) = L*d*x

V (x) = (14 - 2*x) * (8 - 2*x) * x

V (x) = (112 - 28*x - 16*x + 4*x²) * x ⇒ 4*x³ - 44*x² + 112*x

Taking derivatives on both sides of the equation we get:

V' (x) = 12*x² - 88*x + 112

V' (x) = 0 ⇒ 12*x² - 88*x + 112 = 0

A second degree equation, solvin it

3x² - 22*x + 28 = 0

x₁,₂ = [ 22 ± √484 - 336 ] / 6

x₁ = (22 + 12,17) / 6 x₂ = (22 - 12.17) / 6

x₁ = 5.69 We dismiss this solution since it make side 8 - 2x a negative length

x₂ = 9.83/6

x₂ = 1.64

Then x = x₂ = 1.64 in