Stuck,

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.

For the answer to the question above, just to be clear, this is not a high school problem because this is calculus.

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 

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