A golden rectangle is to be constructed such that the longest side is 18 inches long. How long is the other side? (Round your answer to the nearest tenth of an inch.)

The golden ratio satisfies:

a/b=b / (a+b) multiply both sides by (a+b)

(a^2+ab) / b=b multiply both sides by b

a^2+ab=b^2 subtract a^2+ab from both sides

b^2-ab-a^2=0 using the quadratic formula for expediency

b = (a±√ (a^2+4a^2)) / 2 and we know b>0

b = (a+a√5) / 2

b = (a/2) (1+√5)

If we let a=1

b = (1+√5) / 2

So the golden ratio is (1+√5) / 2

Since the longest side is 18in:

(1+√5) / 2=18/s

s (1+√5) = 36

s=36 / (1+√5) in

s≈11.1 in (to nearest tenth of an inch)