Question 120293
Hey! Can I have some help with this equation? 
A box with a square base and no top is to be made from a square piece of cardboard by cutting s = 11 in. squares from each corner and folding up the sides. The box is to hold 704 in3. How big a piece of cardboard is needed?

<pre><b><font size = 5>
We start with an x by x sheet of cardboard like this:

{{{drawing(300,281.25,-5,35,-5,35,
line(0,0,30,0),line(30,0,30,30), line(30,30,0,30),line(0,0,0,30),
locate(31,15,x), locate(15,-1,"x") )}}} 

then we mark off 11x11 squares to cut out of 
the corners, leaving x-22 for the widths of 
the four sides to be folded up.

{{{drawing(300,281.25,-5,35,-5,35,
line(0,0,30,0),line(30,0,30,30), line(30,30,0,30),line(0,0,0,30),
rectangle(0,0,11,11),rectangle(19,0,30,11),rectangle(19,19,30,30),
rectangle(0,19,11,30), locate(31,28,11), locate(31,4,11), locate(31,15,"x-22") 
 )}}}

Now lets cut them out:

{{{drawing(300,281.25,-5,35,-5,35,

line(11,0,11,11),
line(0,11,11,11),
line(0,11,0,19),
line(0,19,11,19),
line(11,19,11,30),
line(11,30,19,30),
line(19,30,19,19),
line(19,19,30,19),
line(30,19,30,11),
line(30,11,19,11),
line(19,11,19,0),
line(19,0,11,0),

 locate(20,28,11), locate(20,4,11), locate(31,15,"x-22") 
 )}}}

Draw in the square base:

{{{drawing(300,281.25,-5,35,-5,35,line(11,0,11,11),
line(0,11,11,11),line(0,11,0,19),line(0,19,11,19),
line(11,19,11,30),line(11,30,19,30),line(19,30,19,19),
line(19,19,30,19),line(30,19,30,11),line(30,11,19,11),
line(19,11,19,0),line(19,0,11,0),
locate(20,28,11), locate(20,4,11), locate(31,15,"x-22"), 
rectangle(11,11,19,19), locate(13,14,"x-22")
 )}}}


So the length and the width will both be
(x-22) and the height will be 11. So the 
volume will be

V = lwh = (x-22)(x-22)(11) = 11(x-22)²

Now we set this = 704

11(x-22)² = 704

Divide both sides by 11

  (x-22)² = 64

Using the square root principle:

   x - 22 = 8

        x = 30

So we needed a 30x30 piece of cardboard.

Edwin</pre>