Final answer:
The area of the large rug is not 3, 6, 9, or 12 times the area of the small rug. In fact, it is 36 times the area of the small rug because the area of a circle is proportional to the square of its radius.
Step-by-step explanation:
The student has provided us with a problem involving the comparison of the areas of two circular rugs. We're given that the radius of the large rug is 3 times the circumference of the small rug. The circumference of the small rug is 2πr, where r is the radius of the small rug. Looking at the large rug, if its radius is 3 times the circumference of the small rug, then its radius is 6πr. To find the areas, we use the formula A = πr² for both rugs and compare them.
For the small rug, its area is πr². For the large rug, given its radius, its area is π(6πr)² = π· 36π²r² = 36π³r². Therefore, the area of the large rug is 36 times the area of the small rug because we are squaring the scale factor, not just multiplying by it. Hence, we can determine that all the given choices A, B, C, and D are incorrect.