Absolutely, let’s go through the steps to solve for [tex]\( C_2 \)[/tex], the circumference of a circle with radius [tex]\( r \)[/tex], given the proportion.
We know the relationship between circumference [tex]\( C \)[/tex] and diameter [tex]\( d \)[/tex] of a circle is given by the formula [tex]\( C = \pi d \)[/tex].
Given:
- [tex]\( d_1 = 1 \)[/tex]
- [tex]\( C_1 = \pi \)[/tex]
- [tex]\( d_2 = 2r \)[/tex]
- We need to solve for [tex]\( C_2 \)[/tex].
Using the provided proportion:
[tex]\[
\frac{C_1}{d_1} = \frac{C_2}{d_2}
\][/tex]
Substitute the known values into the proportion:
[tex]\[
\frac{\pi}{1} = \frac{C_2}{2r}
\][/tex]
Simplify the left-hand side since [tex]\(\frac{\pi}{1} = \pi\)[/tex]:
[tex]\[
\pi = \frac{C_2}{2r}
\][/tex]
To isolate [tex]\( C_2 \)[/tex], multiply both sides of the equation by [tex]\( 2r \)[/tex]:
[tex]\[
C_2 = 2r \pi
\][/tex]
Therefore, the circumference [tex]\( C_2 \)[/tex] of a circle with radius [tex]\( r \)[/tex] is given by:
[tex]\[
C_2 = 2r \pi
\][/tex]
This equation shows that the circumference of a circle is directly proportional to its radius and confirms the well-known formula [tex]\( C = 2\pi r \)[/tex].
