A person has a rectangular piece of cardboard that is 8 cm long and 6 cm wide. He cuts squares of side length $x$ centimetres from each of the corners of the cardboard:
He turns up the sides to form an open box, as shown:
The value for x for which the volume of the box is maximum is?
I know how to get the answer, (that is, by forming an equation for the volume and differentiating to find maximums), and I also know how to find the surface area and hence the volume, that is by simply multiplying the side lengths $(8-2x)$, $(6-2x)$ and $(x)$, but could someone please explain why my original method finding the surface area as $48-4x^2$ be incorrect?
Your calculation is correct: $$2(8-2x)x+2(6-2x)x+(6-2x)(8-2x)=48-4x^2$$