Final answer:
When the area of a circle is described as increasing by a factor of nine, it indicates that the radius has been tripled, thereby making the new area nine times the original area. In this case, the area of the larger circle would be 117 cm².
Step-by-step explanation:
If the area of the original circle is 13 cm², to determine the area of a larger circle resulting from a change in size, one must understand how the area of a circle relates to its radius. The area of a circle is given by the formula A = πr², where A is the area and r is the radius of the circle. If the area of a circle is increased by a factor, it implies that the radius has also changed proportionally since the area is directly proportional to the square of the radius.
When the radius is doubled (increased by a factor of two), the new area becomes (2r)² = 4r², which is four times the original area. Therefore, if the area of a circle is increased by a factor of nine (option b), it suggests that the radius has been tripled, since 3² = 9. Accordingly, the new area is (3r)² = 9r² = 13 cm² * 9 = 117 cm².