To solve the given formula of the circumference of an ellipse for [tex]\( b \)[/tex], follow these detailed steps:
1. Begin with the given circumference formula:
[tex]\[
C = 2 \pi \sqrt{\frac{a^2 + b^2}{2}}
\][/tex]
2. Isolate the square root expression by dividing both sides of the equation by [tex]\( 2 \pi \)[/tex]:
[tex]\[
\frac{C}{2 \pi} = \sqrt{\frac{a^2 + b^2}{2}}
\][/tex]
3. Square both sides to remove the square root:
[tex]\[
\left(\frac{C}{2 \pi}\right)^2 = \frac{a^2 + b^2}{2}
\][/tex]
4. Simplify the left side:
[tex]\[
\frac{C^2}{4 \pi^2} = \frac{a^2 + b^2}{2}
\][/tex]
5. Multiply both sides by 2 to eliminate the fraction on the right:
[tex]\[
\frac{C^2}{2 \pi^2} = a^2 + b^2
\][/tex]
6. Isolate [tex]\( b^2 \)[/tex] by subtracting [tex]\( a^2 \)[/tex] from both sides:
[tex]\[
b^2 = \frac{C^2}{2 \pi^2} - a^2
\][/tex]
7. Finally, solve for [tex]\( b \)[/tex] by taking the square root of both sides:
[tex]\[
b = \sqrt{\frac{C^2}{2 \pi^2} - a^2}
\][/tex]
The correct equation for [tex]\( b \)[/tex] is:
[tex]\[
b = \sqrt{\frac{C^2}{2 \pi^2} - a^2}
\][/tex]
Therefore, the appropriate choice from the provided options is:
[tex]\[
b = \sqrt{\frac{C^2}{2 \pi^2} - a^2}
\][/tex]
