Question 78204
<pre><font size = 4><b>
An artist wants her painting to be in the shape of a golden 
rectangle.  If the length of the painting is 36 inches, 
then what should be the width?

I will assume that since LENGTH comes from the word LONG,
that LENGTH means the LONGER side, and WIDTH is the shorter
side.  Whether that is right or not, I don't know. But
anyway I'll assume it.


 width of golden rectangle        length of golden rectangle
---------------------------- = ----------------------------------
 length of golden rectangle     width+length of golden rectangle

Let w = the width of the golden rectangle:

                           w      36
                         ---- = ------
                          36     w+36

Cross multiply:

                    w(w + 36) = (36)(36)

                     w² + 36w = 1296

              w² + 36w - 1296 = 0

Use the quadratic formula:
                  ______ 
            -b ± <font face = "symbol">Ö</font>b²-4ac
        w = —————————————
                2a 

where a = 1; b = 36; c = -1296

                      _________________ 
             -(36) ± <font face = "symbol">Ö</font>(36)²-4(1)(-1296)
        w = —————————————————————————————
                        2(1) 
                    _________ 
             -36 ± <font face = "symbol">Ö</font>1296+5184
        w = ———————————————————
                    2

                    ____ 
             -36 ± <font face = "symbol">Ö</font>6480
        w = ——————————————
                  2

                    ______ 
             -36 ± <font face = "symbol">Ö</font>1296·5
        w = ———————————————
                   2 

                      _
             -36 ± 36<font face = "symbol">Ö</font>5
        w = ————————————
                  2

                       _
            -36     36<font face = "symbol">Ö</font>5
        w = ———— ± ——————
              2      2
                     _
        w = -18 ± 18<font face = "symbol">Ö</font>5 
                           _
Using the +, w = -18 + 18 <font face = "symbol">Ö</font>5, which
is one solution and equals about 22.25
                          _ 
Using the -, w = -18 - 18<font face = "symbol">Ö</font>5, which
is one solution and equals about -58.25

We discard the negative answer, and the answer is

Width = 22.25 inches, rounded to the nearest hundredth.

Edwin</pre>