You are given that the perimeter of the rectangle is 66 cm;  hence, the sum of the length and the width is one half of that:

x + y = 33,  where x is the length, and y is the width.


Then the average of the length and the width is one half of 22, i.e. 16.5.


It is clear that the values of x and y are remoted at the same value/distance "u" from the average of 16.5, so we can write

x = 16.5 + u,
y = 16.5 - u.


Then the area is  xy = (16.5+u)*(16.5-u) = , and it is equal to 270, according to the condition.

Hence,   = 270,  which gives   = 272.25 - 270 = 2.25,  and then  u =  = 1.5.


Thus the length is  x = 16.5 + u = 16.5 + 1.5 = 18,

and  the width  is  x = 16.5 - u = 16.5 - 1.5 = 15.


Answer.  The dimensions of the rectangle are  15 cm  and  18 cm.