Question

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?

Answer 1

1.64 in

Explanation:

Answer 2

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

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