Question

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.

Answer 1

V = x · y²
2 x + y = 3
y = 3 - 2 x
V = x · ( 3 - 2 x )² = x · ( 9 - 12 x + 4 x² ) = 9 x - 12 x² + 4 x³
V ` = 9 - 24 x + 12 x² = 3 ( 4 x² - 8 x + 3 ) = 3 ( 4 x² - 6 x - 2 x + 3 ) =
= 3 [ 2 x ( 2 x - 3 ) - ( 2 x - 3 ) ] = 3 ( 2 x - 3 ) ( 2 x - 1 )
The largest volume is when: V ` = 0, so:
2 x - 3 = 0  
2 x = 3               
x = 1.5 ( which is incorrect, because : y = 0 ) 
or: 2 x - 1 = 0
2 x = 1
x = 0.5,  y = 3 - 2 · 0.5 = 3 - 1 = 2
V max = 0.5 · 2² = 0.5 · 4 = 2 ft³