A box is to be made out of a 12 cm by 18 cm piece of cardboard. Squares of side length x 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 V of the box as a function of x

B) Give the domain of V in interval notation.

C) Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W≤L).

D) What is the maximum volume of the box?

I was able to get part A which is: (12 - 2x)(18 - 2x)(x) but I'm struggling with the rest.

Question

A box is to be made out of a 12 cm by 18 cm piece of cardboard. Squares of side length x 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 V of the box as a function of x

B) Give the domain of V in interval notation.

C) Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W≤L).

D) What is the maximum volume of the box?

I was able to get part A which is:  (12 - 2x)(18 - 2x)(x) but I'm struggling with the rest.

radar · Answer

Due to the actual dimensions of the cardboard the 12 cm dimension will limit x to be less than 6.  dimensions are positive so the domain of V is restricted to 0<x<6.

radar · Answer

You can see if x became 6 then the 12 cm would be split in half and there would be nothing left to make a base for the box.

radar · Answer

Are you following with understanding?

radar · Answer

You need to continue and do the multiplication getting: V=4x^3 - 60x^2 = 216x for part a)