Answer :
Answer:
[tex]b = \sqrt{\frac{C^{2} }{2(\pi )^{2} } - a^{2}}[/tex]
Step-by-step explanation:
Given - The circumference of the ellipse approximated by [tex]C = 2\pi \sqrt{\frac{a^{2} + b^{2} }{2} }[/tex]where 2a and 2b are the lengths of 2 the axes of the ellipse.
To find - Which equation is the result of solving the formula of the circumference for b ?
Solution -
[tex]C = 2\pi \sqrt{\frac{a^{2} + b^{2} }{2} }\\\frac{C}{2\pi } = \sqrt{\frac{a^{2} + b^{2} }{2} }[/tex]
Squaring Both sides, we get
[tex][\frac{C}{2\pi }]^{2} = [\sqrt{\frac{a^{2} + b^{2} }{2} }]^{2} \\\frac{C^{2} }{(2\pi)^{2} } = {\frac{a^{2} + b^{2} }{2} }\\2\frac{C^{2} }{4(\pi)^{2} } = {{a^{2} + b^{2} }[/tex]
[tex]\frac{C^{2} }{2(\pi )^{2} } = a^{2} + b^{2} \\\frac{C^{2} }{2(\pi )^{2} } - a^{2} = b^{2} \\\sqrt{\frac{C^{2} }{2(\pi )^{2} } - a^{2}} = b[/tex]
∴ we get
[tex]b = \sqrt{\frac{C^{2} }{2(\pi )^{2} } - a^{2}}[/tex]