Thursday, November 3rd

Problem
Can an arc of a parabola inside a circle of radius 1 have a length greater than 4? 




Solution
No. Without loss of generality let $y=ax^2$ be the parabola. After a bit, one can be convinced that in order to maximize the arc length inside the circle, it has to have its center at $(0,1)$. 


By finding the arc length as a function of $a$ and applying the fundamental theorem of calculus, one can see that the arc length of the parabola inside the circle is an increasing function of $a$. Since for $a=0$ the arc length is 0 and for $a=\infty$ the arc length is 4, there is no such solution for the problem.