Does max and min area work the same way in ellipses as in quadrilaterals?

Question

Max area of a quadrilateral with a certain perimeter isn't a square most of the time. Is max area of an ellipse with a certain circumference not usually a perfect circle? Is it possible to use a quadratic equation to solve the values of the extreme radii (what's the proper term?) in this case? What about a sphere? Doesn't a perfect sphere have optimal volume? Is area that much different than volume? Why? ¦D This is what my parents put up with!

Answer 1

Circles and spheres and so on, unless I'm very mistaken, are going to surround the greatest volume for their surface areas. (In the case of a circle, that would be the circumference.)

For the specific case of ellipses, google on ellipse circumference, or just go to the Wikipedia article on ellipses. While there's no closed form formula for an ellipse's area, the approximation they give sure looks like it would be maximized when a and b are equal. (I didn't actually check.)

Answer 2

i'm going to provide you a star for this exciting takeoff of yet another published situation. the respond, of direction, is only ?, yet i do no longer prefer to destroy the plot for the different individuals. enable's only say that i'm "astounded" to locate that the portion of the ellipses of e(0) and e(a million) are the two ?, whilst "laboriously" worked out. Vikram gave a sturdy answer. yet in any different case to do it somewhat is to set up the equation to: (F(n) x + F(n+a million) y)² / a million² +(F(n+2) x + F(n+3) y)² / a million² = a million wherein the section is (a million)(a million)? / |F(n)F{n+3) - F(n+a million)F(n+2)|. Then it somewhat is a controversy of proving that the Jacobian, the term interior the denominator, is a million. we can borrow VIkram's evidence for that.