SOLUTION: 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?

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Question 78204: 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?
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
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?
--------
If a rectangle has length a and width b and the larger demention is "a",
the golden ratio is (a+b)/a =a/b
---------
Then larger/smaller = (1+sqrt5)2 = 1.6180339887...
---------
Your problem:
If 36 is the larger dimension for the picture you get:
36/smaller = 1.6180.../1
smaller = 36/1.6180=22.249...
-----------
If 36 is the smaller dimension for the picture you get:
larger/36 = 1.6180
larger = 1.6180,,,*36 = 58.24922....
==========
Cheers,
Stan H.

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

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 ± Öb²-4ac
        w = —————————————
                2a 

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

                      _________________ 
             -(36) ± Ö(36)²-4(1)(-1296)
        w = —————————————————————————————
                        2(1) 
                    _________ 
             -36 ± Ö1296+5184
        w = ———————————————————
                    2

                    ____ 
             -36 ± Ö6480
        w = ——————————————
                  2

                    ______ 
             -36 ± Ö1296·5
        w = ———————————————
                   2 

                      _
             -36 ± 36Ö5
        w = ————————————
                  2

                       _
            -36     36Ö5
        w = ———— ± ——————
              2      2
                     _
        w = -18 ± 18Ö5 
                           _
Using the +, w = -18 + 18 Ö5, which
is one solution and equals about 22.25
                          _ 
Using the -, w = -18 - 18Ö5, 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