Circumference C=2(a+b)

According to Cauchy inequality:

[(a/2)^2+b^2](4+ 1)>； =(a+b)^2=(C/2)^2

So the maximum circumference is (2 root numbers 5) * r.

If and only if a=4b, the above equal sign holds.

So when the perimeter of this rectangle is the largest, a/b=4.