Question

The circumference of the ellipse approximate. Which equation is the result of solving the formula of the circumference for b?

c=2\pi \sqrt{a^2+b^2} /2

Answer 1

`b = \sqrt{(C^(2) )/(2(\pi )^(2) ) - a^(2)}`

Explanation:

Given - The circumference of the ellipse approximated by
`C = 2\pi \sqrt{(a^(2) + b^(2) )/(2) }`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 -

`C = 2\pi \sqrt{(a^(2) + b^(2) )/(2) }\\(C)/(2\pi ) = \sqrt{(a^(2) + b^(2) )/(2) }`

Squaring Both sides, we get

`[(C)/(2\pi )]^(2) = [\sqrt{(a^(2) + b^(2) )/(2) }]^(2) \\(C^(2) )/((2\pi)^(2) ) = {(a^(2) + b^(2) )/(2) }\\2(C^(2) )/(4(\pi)^(2) ) = {{a^(2) + b^(2) }`

`(C^(2) )/(2(\pi )^(2) ) = a^(2) + b^(2) \\(C^(2) )/(2(\pi )^(2) ) - a^(2) = b^(2) \\\sqrt{(C^(2) )/(2(\pi )^(2) ) - a^(2)} = b`

∴ we get

`b = \sqrt{(C^(2) )/(2(\pi )^(2) ) - a^(2)}`