Question

multiple - choice 20 points circle a has a radius of 3n and circle b has a radius of 129n, where n is a positive constant. the area of circle b is how many times the area of circle a? 129 86 43 1849

Explanation:

Step1: Recall the area - formula of a circle

The area formula of a circle is $A = \pi r^{2}$, where $r$ is the radius of the circle.
For circle $A$ with radius $r_{A}=3n$, its area $A_{A}=\pi(3n)^{2}=9\pi n^{2}$.
For circle $B$ with radius $r_{B} = 129n$, its area $A_{B}=\pi(129n)^{2}=16641\pi n^{2}$.

Step2: Calculate the ratio of the areas

We want to find out how many times the area of circle $B$ is that of circle $A$. Let the ratio be $k=\frac{A_{B}}{A_{A}}$.
Substitute the expressions for $A_{A}$ and $A_{B}$ into the ratio formula:
$k=\frac{16641\pi n^{2}}{9\pi n^{2}}$.
Since $n$ is a non - zero positive constant (because it represents a radius - related quantity), the $\pi n^{2}$ terms in the numerator and denominator cancel out, and $k = 1849$.

Answer:

1849