Verify my setup of this applied max/min problem

[hank]

I'm pretty sure the book is wrong on the answer for this one (surprise, surprise this books sucks)

Anyway, the problem:

A sheet of cardboard 12 in square is used to make an open box by cutting squares of equal size from the four corners and folding up the sides. What size squares should be cut to obtain a box with the largest possible volume?

Alright, so I have a square. First thing I want to do is find out what the sides of the square are.

y^2 = 12
y = 2 * sqrt 3

However, there are going to be 2 equal sized squares cut from each side. Lets call their size x. Therefore, each side is equal to:

2 * sqrt(3) - 2x.

Ok, now I'm looking for the biggest volume of the box. Therefore, the volume is going to equal:

V = x(2 * sqrt(3) - 2x)^2 //Height is x, and the sides are the same as for the square.

Now, I need to find d/dx[V].

Next, I set v'(x) to zero and solve for x.

One of those x values are going to be the size of my squares, right?

 

[galactus]

Let x=the length of the side of a square.

V=volume

\(\displaystyle V=(12-2x)(12-2x)x=4x^{3}-48x^{2}+144x, \;\ 0\leq{x}\leq{6}\)

Differentiate:

\(\displaystyle 12x^{2}-96x+144=0\)

Now, set to 0 and solve for x. You'll get 2 answers. One will be extraneous because it'll fall outside or on the interval [0,6].

 

[hank]

Ok, that's what threw me off.

I thought that 12 in squared meant that the total was 12 in squared, not 12 in to a side.

Thx.