Question 1207601
<pre>
An open box is to be constructed from a square piece of sheet metal by removing a square of side 1 foot from each corner and turning up the edges. If the box is to hold 5 cubic feet, what should be the dimension of the sheet metal? 

Note: This is a popular problem found in many textbooks but I haven't found one decent explanation. Can someone solve this application one step at a time?

Since the dimensions of the sheet metal are being sought, we let the width, which is also the same measure as
the length (the sheet metal is SQUARE), be W
As a 1-foot square piece is removed from each end of the sheet metal, the width and the length of the base/bottom
of the box will be W - 2, each.
The sides/height of the box will be formed from the 1-foot cut-outs 
So, as the open box will have a volume (Length * Width * Height, or LWH) of 5 cubic feet, we get:
L*W*H = 5. Now, with W (width) = W - 2 (same measure as the length, or L), LWH = 5 becomes:
(W - 2)(W - 2)(1) = 5
       <font color = red><font size = 4><b>(W - 2)<sup>2</sup> = 5</font></font></b> 
   <font color = red><font size = 4><b>W<sup>2</sup> - 4W + 4 = 5</font></font></b> 
   <font color = red><font size = 4><b>W<sup>2</sup> - 4W - 1 = 0</font></font></b>
As the above quadratic cannot be factored with INTEGERS, solutions can be derived by using the quadratic equation 
formula or completing the square. Using the former, we get: {{{matrix(1,3, W, "=", (- b +- sqrt(b^2 - 4ac))/(2a))}}}                                     
                                {{{matrix(1,3, W, "=", (- - 4 +- sqrt((- 4)^2 - 4(1)(- 1)))/(2(1)))}}}, with {{{matrix(3,3, a, "=", 1, b, "=", - 4, c, "=", - 1)}}}
                                {{{matrix(7,3, W, "=", (4 +- sqrt(16 + 4))/2, W, "=", (4 +- sqrt(20))/2, W, "=", (4 +- sqrt(4  * 5))/2, W, "=", (4 +- sqrt(4)*sqrt(5))/2, W, "=", (4 +- 2*sqrt(5))/2, W, "=", 2(2 +- sqrt(5))/2, W, "=", cross(2)(2 +- sqrt(5))/cross(2))}}}

Width/Length of sheet metal, or {{{highlight_green(matrix(1,7, W, "=", 2 + sqrt(5), "=", 4.236, feet, "(approximately)"))}}}   or  {{{matrix(1,3, W, "=", 2 - sqrt(5))}}} <=== IGNORE/REJECT as
                                                                                   length|width CANNOT be negative (<0) 

  Dimensions of sheet metal: {{{highlight_green(matrix(1,11, 2 + sqrt(5), by, 2 + sqrt(5), ",", or, approximately, 4.236, feet, by, 4.236, feet))}}}</pre>