A box is to be made out of a 6 cm by 18 cm piece of cardboard. Squares of side length [math] cm will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top.

(a) Express the volume [math] of the box as a function of [math].

[math] [math]

(b) Give the domain of [math] in interval notation. (Use the fact that length and volume must be positive.)

(c) Find the length [math], width [math], and height [math] of the resulting box that maximizes the volume. (Assume that [math]).

[math] = cm

[math] = cm

[math] = cm

(d) The maximum volume of the box is [math].