To solve for [tex]\( C_2 \)[/tex], the circumference of any circle with radius [tex]\( r \)[/tex], we should follow these steps:
1. Understand the given proportion: The proportion [tex]\( \frac{C_1}{d_1} = \frac{C_2}{d_2} \)[/tex] relates the circumference and diameter of two circles. Here, the diameter of the first circle [tex]\( d_1 = 1 \)[/tex] and its circumference [tex]\( C_1 = \pi \)[/tex]. The second circle has a diameter [tex]\( d_2 = 2r \)[/tex] and we need to find its circumference [tex]\( C_2 \)[/tex].
2. Substitute the given values:
[tex]\[
\frac{C_1}{d_1} = \frac{C_2}{d_2}
\][/tex]
Substituting [tex]\( C_1 = \pi \)[/tex], [tex]\( d_1 = 1 \)[/tex], and [tex]\( d_2 = 2r \)[/tex]:
[tex]\[
\frac{\pi}{1} = \frac{C_2}{2r}
\][/tex]
3. Solve for [tex]\( C_2 \)[/tex]:
[tex]\[
\frac{\pi}{1} = \frac{C_2}{2r}
\][/tex]
Multiply both sides by [tex]\( 2r \)[/tex] to isolate [tex]\( C_2 \)[/tex]:
[tex]\[
\pi \cdot 2r = C_2
\][/tex]
Simplify:
[tex]\[
C_2 = 2r\pi
\][/tex]
Therefore, the correct solution is:
[tex]\[
\boxed{\text{Because } \frac{\pi}{1} = \frac{C_2}{2r}, \text{ then } C_2 = 2r\pi.}
\][/tex]
