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Geometry and Trigonometry / Area and volume Difficulty: Hard

Circle $A$ has a radius of $3 n$ and circle $B$ has a radius of $129 n$ , where $n$ is a positive constant. The area of circle $B$ is how many times the area of circle $A$ ?

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Explanation

Choice D is correct. The area of a circle can be found by using the formula $A = π r^{2}$ , where $A$ is the area and $r$ is the radius of the circle. It’s given that the radius of circle A is $3 n$ . Substituting this value for $r$ into the formula $A = π r^{2}$ gives $upper A equals pi left parenthesis 3 n right parenthesis squared$ , or $9 pi n squared$ . It’s also given that the radius of circle B is $129 n$ . Substituting this value for $r$ into the formula $A = π r^{2}$ gives $upper A equals pi left parenthesis 129 n right parenthesis squared$ , or $16,641 pi n squared$ . Dividing the area of circle B by the area of circle A gives $StartFraction 16,641 pi n squared Over 9 pi n squared EndFraction$ , which simplifies to $1,849$ . Therefore, the area of circle B is $1,849$ times the area of circle A.

Choice A is incorrect. This is how many times greater the radius of circle B is than the radius of circle A.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the coefficient on the term that describes the radius of circle B.