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