Final answer:
The ratio of the circumference of circle B is 0.9 to that of circle A, leading to a radius ratio of 0.9. The area ratio is the square of the radius ratio, which results in the area ratio of circle A to circle B being 100:81.
Step-by-step explanation:
The question involves the circumference and area of circles, which are fundamental concepts in geometry, a branch of mathematics. We use the relationship between a circle's circumference and its radius (C = 2πr) to understand the relationship between the areas of the circles. If the circumference of circle B is 90% of the circumference of circle A, then the ratio of their radii is also 0.9 (since circumference is directly proportional to radius). The area of a circle is given by A = πr². Therefore, the ratio of the areas of circles A and B can be calculated using the squares of the ratio of their radii.
Knowing that the circumference of circle B is 90% of that of circle A, we can write:
- Circumference of A = 2πrA
- Circumference of B = 2π× 0.9rA = 2πrB
Hence, rB = 0.9rA. Therefore, the ratio of the areas is:
Area of A / Area of B = πrA² / π(rB)² = (rA/rB)² = (1/0.9)² = (10/9)² = 100/81.
Thus, the ratio of the area of circle A to the area of circle B is 100:81.