The question follows like this:
Show that the formula for the T.S.A, A (in square meters) of the box is A=-6x^2 + 12x
The information that was given is:
8x + 4y=12
also
y= 3 - 2x
What you have written makes no sense at all. You ask for the total surface area of a box but say nothing about how "x" and "y" are related to the box. Perhaps x and y are two of the dimensions (length and height, say) but what is the third dimension (width)? Also, you say, "8x+ 4y= 12 also y= 3- 2x" but those are, in fact, equations of same line in the xy-plane.
I would like to help but I really have no idea what the problem really is.
If the dimensions of the rectangular solid are x, y, and z, then the total surface area is 2xy+ 2yz+ 2xz. If y= 3- 2x, then the total surface area is $\displaystyle 2x(3- 2x)+ 2(3- 2x)z+ 2xz= 6x- 4x^2+ 6z- 4xz+ 2xz= 6z- 4xz+ 6x- 4x^2$. In order for that to be equal to $\displaystyle 12x- 6x^2$ we must have $\displaystyle 6z- 4xz+ 5x- 4x^2= 12x- 6x^2$ which reduces to $\displaystyle (6- 4x)z= 7x- 2x^2$ or $\displaystyle z= \frac{7x- 2x^2}{6- 4x}$. Was there any condition like that in your problem?
NO. Here is the sum in full is.
A piece of wire 12m long, is bent to form a rectangular frame. The lenght is y m and the breadth and the height are x m. We can deduce that: 8x + 4y = 12
If you make y the subject it becomes y= 3 - 2x
The whole frame is coverd with cardboard to make a box.
Show that the formuls for the T.S.A, A (in square metres) of the box is A= -6x^2 + 12x.
"The breadth and the height are x m". Yes, you are given the z component! It is the same as the x component. The total surface area is, as I said before, 2xy+ 2yz+ 2xy and, because we now know that the problem said that x= z. 2xy+ 2yz+ 2xy= 2xy+ 2xy+ 2x^2= 4xy+ 2x^2. Now, put y= 3- 2x into that.
Surface Area of a rectangular geometry would be given as Area = 2(wh + lw + lh). To get the easier conversions on area or surface unit you may check out with the
Area Unit Converter, comprising of imperial conversion units.