From each corner of a square piece of cardboard, a square with sides of 3cm is removed. The edges are then up to form an open box. If the box is to hold 243cm^3, what are the dimensions of the original piece of cardboard?

original piece ---- x cm by x cm

base is (x-6) by (x-6)

x(x-6)^2 = 243
x^3 - 12x^2 + 36x - 243 = 0

I don't know what method you have to solve a cubic. Since this one does not factor it becomes complicated. Newton's Method is one algorithm.
I sent it through Wolfram and got 10.754

http://www.wolframalpha.com/input/?i=x%5E3+-+12x%5E2+%2B+36x+-+243+%3D+0

To find the dimensions of the original piece of cardboard, we need to follow a step-by-step process.

Let's start by analyzing the information provided in the question:

1. A square piece of cardboard is given.
2. From each corner of the cardboard, a smaller square with sides of 3cm is removed.
3. The remaining edges are folded up to form an open box.
4. The volume of this box is specified as 243cm^3.

Now, let's find the dimensions of the original piece of cardboard.

Step 1: Calculate the volume of the original piece of cardboard
Since the volume of the box is 243cm^3, we can conclude that this is also the volume of the piece of cardboard before any squares were removed.

Step 2: Calculate the volume of the removed squares
Each removed square has sides of 3cm, so their volume can be calculated as (3cm)^2 = 9cm^2.

There are four corners, so a total of 4 squares are removed. Thus, the total volume of the removed squares is 4 * 9cm^2 = 36cm^2.

Step 3: Subtract the volume of the removed squares from the volume of the original piece
To find the volume of the original piece, we subtract the volume of the removed squares from the volume of the box:
Original volume = 243cm^3 + volume of the removed squares = 243cm^3 + 36cm^2 = 279cm^2.

Step 4: Calculate the dimensions of the original piece
Since we have a square piece of cardboard, the length and width are equal. Let's call the length/width of the original piece "x."

The formula for the volume of a cube is V = x^3. In this case, the volume is 279cm^2. So we have the equation:
x^3 = 279cm^2.

To find the value of x, we can take the cube root of both sides of the equation:
x = ∛279cm^2 ≈ 6.24cm.

Therefore, the original piece of cardboard is approximately 6.24cm by 6.24cm in dimensions.