Question

the circumference of an ellipse is approximated by (c = 2pisqrt{\frac{a^{2}+b^{2}}{2}}) where (2a) and (2b) are the lengths of the axes of the ellipse. which equation is the result of solving the formula for (b)?

Explanation:

Step1: Start with the given formula for the circumference of an ellipse

$C = 2\pi\sqrt{\frac{a^{2}+b^{2}}{2}}$. We want to solve for $b$. First, divide both sides by $2\pi$.
$\frac{C}{2\pi}=\sqrt{\frac{a^{2}+b^{2}}{2}}$

Step2: Square both sides to get rid of the square - root

$(\frac{C}{2\pi})^2=\frac{a^{2}+b^{2}}{2}$

Step3: Multiply both sides by 2

$2\times(\frac{C}{2\pi})^2=a^{2}+b^{2}$

Step4: Subtract $a^{2}$ from both sides

$b^{2}=2\times(\frac{C}{2\pi})^2 - a^{2}$

Step5: Take the square - root of both sides

$b=\sqrt{2\times\frac{C^{2}}{4\pi^{2}}-a^{2}}=\sqrt{\frac{C^{2}}{2\pi^{2}}-a^{2}}$

Answer:

$b = \sqrt{\frac{C^{2}}{2\pi^{2}}-a^{2}}$