The area in square inches is 144*96=13824. The area of a rectangle is 22*16=352. The maximum number of rectangles is 13824/352=39.273. But if we cannot use part of a rectangle the maximum is 39. 40 rectangles would allow the area to be filled completely but it would be necessary to cut up some rectangles to fit.
Since 96 is exactly divisible by 16, the 8' side of the area will allow 6 rectangles to be placed with their 16" edges forming an 8' edge. The 22" edge will not cover 12', but 6 rectangles will cover 6*22=132 out of 144 inches, leaving 12 inches short. The size of this remaining area is 12" by 96". So far we've used 6*6=36 rectangles, and the maximum is 40. If 3 more rectangles are split into two, we have 6 rectangles of size 11" by 16" which will partly fill the 12" gap along the 8' edge of the area, leaving only 1" by 8' to fill. Now we have used 39 rectangles, 3 of which were divided in half. We would need to cut 6 1" by 16" strips from another 22" by 16" rectangle to fill the remaining area.
So we used 36 complete rectangles, plus another 3 cut in half, plus part of another cut into 6 1" strips, making 40 rectangles in all completely covering the area, and part of a rectangle unused.