Orlando is making a design for a logo he begins with a square measuring 22 inches on a side. The second square has a side length of 17.6 inches and the third square has a side length of 14.08. Which square will be the first square with a side length of less than 10 inches

Given that the first square measures 22 inches and the second square measures 17.6 inches. The ratio of the length of the second square to the length of the first square is 17.6 / 22 = 0.8

The ratio of the length of the third square to the length of the second square is 14.08 / 17.6 = 0.8

The scenario gives a decreasing geometric sequence with a common ratio of 0.8

Thus, the length of the fourth square = 14.08 x 0.8 = 11.264 inches and the length of the fifth square is 11.264 x 0.8 = 9.0112 inches.

Therefore, the first square with a side length of less than 10 inches is the fifth square.