Maximum volume

Question

From a thin rectangular piece of cardboard 8 inches on a side, small squares are cut out of the corners. The sides are then folded up to form a box. What size squares should be cut out to maximize the volume of the box? What is the maximum volume of the box?

Answer 1

Let x = length of a side of a cut out square

 

V = volume of box = x(8 - 2x)(8 - 2x)

 

V = x(64 - 32x + 4x²)

 

V = 4x³ - 32x² + 64x

 

V' = 12x² - 64x + 64

    = 4(3x² - 16x + 16)

    = 4(3x - 4)(x - 4)

 

V' = 0 if x = 4/3 or x = 4

 

x ≠ 4 since the cardboard is only 8" on a side, so cutting 4" would cut the cardboard sheet in half.  

 

When 0 < x < 4/3, V'> 0 and when 4/3 < x < 4, V' < 0

 

So, the volume is maximized when x = 4/3