You cut square corners with side lengths that are whole numbers from a piece of cardboard with dimensions 20 inches by 30 inches. You then fold the cardboard to create a box with no lid. Which of the following dimensions will give you the greatest volume?
A. 12 in. by 22 in. by 4 in.
B. 10 in. by 20 in. by 5 in.
C. 14 in. by 24 in. by 2 in.
D. 10 in. by 24 in. by 6 in.

Suppose the squares you cut have side length x, [ then we have:
Length of box base: 20 - 2x
Width of box base: 30 - 2x
Height of box: x
Volume of box:
V = x(20 - 2x)(30 - 2x)
V = x(600 - 40x - 60x + 4x^2)
V = x(600 - 100x + 4x^2)
V = 600x - 100x^2 + 4x^3
dV/dx = 600 - 200x + 12x^2
Set that to zero:
12x^2 - 200x + 600 = 0
x = (-(-200) +/- sqrt((-200)^2 - 4(12)(600))) / (2*12)
x = (200 +/-

sqrt(40000 - 28800)) / 24
x = (200 +/- sqrt(11200)) / 24
x = (200 +/- sqrt(11200)) / 24
x =~ 12.74 or 3.92
Since you want whole numbers, x = 13 or 4
If x = 13, V = -312, but a negative number doesn't make sense, so we'll ignore that
If x = 4, V = 1056, so that's the maximum volume
The dimensions are:
Length of box base: 12
Width of box base: 22
Height of box: 4
By the way, this is calculus, not grade 8 math! ]

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