Question

The sides of a square are 27 inches in length. New squares are formed by dividing the original square into nine squares. The center square is then shaded (see figure ). If this process is repeated a t

Answer 1

The total shaded area after an infinite number of iterations will be
`(5)/(27)`of the original square's area.

Step-by-step explanation:

In the initial square, dividing it into nine smaller squares creates eight identical squares around the center square. The area of the original square is
`27^(2)` = 729 square inches. Among the nine smaller squares formed, the center square is shaded. After the first iteration, the area of the shaded region is
`27^(2)` - (3× 3× 27) = 729 - 243 = 486 square inches.

After each subsequent iteration, eight equally sized squares surround the center square, and the total shaded area within the larger square reduces by
`(4)/(9)` of the previous shaded area. This forms a geometric series where the sum of an infinite series can be calculated using the formula for an infinite geometric series:
`S=(a)/(1-r')`, where
`s` is the sum of the series,
`a` is the first term, and
`r` is the common ratio.

In this case, a = 486 square inches (the shaded area after the first iteration) and r =
`(4)/(9)` (the ratio by which the shaded area reduces with each iteration). Substituting these values into the formula gives S=
`(486)/(1-(4)/(9) ) =(486)/((5)/(9) ) =(486(9))/(5)` = 874.8 square inches. Thus, the total shaded area after an infinite number of iterations will be
`(5)/(27)` of the original square's area, which is
`(5)/(27)` × 729 = 135 square inches.

Answer 2

Marta's larger square, which has dimensions twice that of the original 4-inch square, results in an area that is four times greater. The original square's area is 16 square inches, while the larger square's area is 64 square inches.

Step-by-step explanation:

The question mentions that Marta has an original square with a side length of 4 inches. She also has a similar square whose dimensions are twice that of the first square.

To find the area of the larger square, we use the scale factor, which is 2 in this case. This means that each side of the larger square is 4 inches × 2, resulting in side lengths of 8 inches.

Now, to compare the areas of the two squares, we use the formula for the area of a square, which is side length squared (A = s²). Therefore, the area of the smaller square is 4 inches × 4 inches = 16 square inches. The area of the larger square is 8 inches × 8 inches = 64 square inches.

Comparing the two areas, the larger square's area is 64 square inches which is four times the area of the smaller square (16 square inches). This illustrates that when scaling up the dimensions of a figure by a factor of 2, the area increases by a factor of 2², or 4.