Let #x and y# be the length and width of the rectangle in inches.
Then , perimeter of the rectangle is #p=2(x+y)= 23# or
#x+y= 23/2 or x+y= 11.5 (1)# and the area of the rectangle is
#a=x*y =30.66 :. 4xy= 4*30.66= 122.64#
#(x-y)^2= (x+y)^2-4xy or (x-y)^2= 11.5^2-122.64 # or
#(x-y)^2= 11.5^2-122.64= 9.61 :. (x-y)= sqrt 9.61# or
#x- y =3.1 (2)# Adding equation (1) and equation (2) we
get , #2x=11.5+3.1 or 2x= 14.6 or x= 7.3 # inches
#y = 11.5-7.3 or y = 4.2 # inches.
#7.3 and 4.2# inches are length and width of the rectangle. [Ans]