SOLUTION: This is a challenge problem on my homework and I don't completely understand it. Would someone please help me? A box with a square bottom is to be made from a 12 inch square

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Question 703484: This is a challenge problem on my homework and I don't completely understand it. Would someone please help me?

A box with a square bottom is to be made from a 12 inch square piece of cardboard. Squares with the side x are cut out of the corners, and the sides are folded up. Express the volume and the surface area of the outside of the box as polynomials.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
This problem keeps coming up and I am always re-drawing the picture.

As you cut squares out of each corner (I made them red),
each square measuring x inches to a side,
you get four x-inch wide flaps to fold up around the (green) square bottom.
The side of that green square is 12-2x inches long.
The surface area of the green square is %2812-2x%29%5E2=144-48x%2B4x%5E2 square inches.
The flaps/sides will be x inches by 12-2x inches,
with a total surface area of 4x%2812-2x%29=48x-8x%5E2 square inches.
The surface area of the box (just four sides and bottom, because we have no top/lid) is
A%28x%29=4x%5E2-48x%2B144%2B48x-8x%5E2 square inches, so
A%28x%29=-4x%5E2%2B144
The box will be x-inches high,
with a base surface of 144-24x%2Bx%5E2 square inches (as calculated above),
and a volume of
Volume+=x%2A%28144-48x%2B4x%5E2%29 --> Volume+=4x%5E3-48x%5E2%2B144x

NOTE:
I know I make stupid arithmetic mistakes, so don't trust my calculations.