Final answer:
The area of Circle B, with a diameter three times that of Circle A, will be 4500 in², obtained by tripling the radius of Circle A and using the formula A = πr², with area scaling by the square of the tripling factor.
Step-by-step explanation:
To find the area of Circle B, whose diameter is three times the diameter of Circle A, we must first understand how the area of a circle is related to its diameter. The area (A) of a circle is calculated using the formula A = πr², where r is the radius of the circle. Knowing that the radius is half of the diameter, we can establish a relationship between the areas of the two circles.
Circle A has an area of 500 in², which we can represent as A = πr² = 500. To find the radius of Circle A, we rearrange the formula to solve for r: r = √(500/π). Once we have the radius of Circle A, we can find the radius of Circle B by tripling it, since the diameter of Circle B is three times that of Circle A.
With the radius of Circle B calculated, we use it to find the area of Circle B with the same area formula: A = πr². Because each linear dimension (diameter and radius) is tripled, the area of Circle B will be nine times the area of Circle A, as the area scales with the square of the radius.
Therefore, the area of Circle B will be 9 × 500 in² = 4500 in².