The length of the box is (30-2x) because you are removing x from both ends of the box

The width of the box is (6-2x) because you are removing x from both ends of the box

the height is x because when you fold up the flaps you've created they will come x high

Volume of a rectangular prism is length x width x height

V=lwh

plug in values

V=(30-2x)(6-2x)x

Expand if needed:

V=(180-60x-12x+4x^{2})

V=180x-72x^{2}+4x^{3}

Domain is the possible values for x. Since this is a real 3D shape, l,w,h cannot be negative. The smallest x could be and not result in a negative value is x=0: l=30, w=6, h=0 although this wouldn't create a box just a flat piece of cardboard.

The largest would be what would make either w=0 or l=0

6-2x=0

6=2x

3=x

30-2x=0

30=2x

15=x

Since x=3 is smaller, this is the largest the x value can be since any bigger width would be negative.

Domain would be:

[0,3] <--- if you count a zero volume box to be valid

(0,3) <--- if you only count a box where V>0