Final answer:
The circumference of circle A is 3 times smaller than the circumference of Circle B because the radius of Circle B is 3 times that of Circle A and circumference is directly proportional to the radius.
Step-by-step explanation:
The problem involves comparing the area and the circumference of two circles, Circle A and Circle B, where the area of Circle A is 9 times smaller than the area of Circle B. To find out by how much the circumference of Circle A is smaller than that of Circle B, we first recognize the relationship between the area and the radius of a circle, which is A = πr².
Since the area of Circle A is 9 times smaller than Circle B, if we consider the area of Circle B to be A_B = 9A_A, and the radius of Circle B is r_B and the radius of Circle A is r_A, then the square of the radius of Circle B will be 9 times the square of the radius of Circle A (r_B² = 9r_A²). Taking the square root of both sides, we get r_B = 3r_A, which tells us that the radius of Circle B is 3 times that of Circle A.
Now, the circumference of a circle is given by the formula C = 2πr. Since the radius of Circle B is 3 times that of Circle A, the circumference of Circle B will also be 3 times greater than the circumference of Circle A. Thus, the circumference is directly proportional to the radius.
Therefore, the circumference of Circle A is 3 times smaller than the circumference of Circle B.