Arc \(AC\) is a quarter-circle with center \(B\). The shaded region \(ABC\) is "rolled" along a straight board \(PQ\) until it reaches its original orientation for the first time with point \(B\) landing at point \(B^{\prime}\). If \(BC = \frac{2}{\pi}\) cm, what is the length of the path that point \(B\) travels? Express your answer in simplest form.

TheGreatestOofman Apr 6, 2021

#1**+2 **

First it travels one more radius, then the circumference. then one more radius. So, we have $\frac{2}{\pi}+\frac{\pi*2*\frac{2}{\pi}}{4}+\frac{2}{\pi}=\frac{4}{\pi}+\frac{2*2}{4}=\boxed{1+\frac{4}{\pi}}$

RiemannIntegralzzz Apr 6, 2021

#2**0 **

Sorry, that is wrong because the point B is supposed to travel on a curved path, not just including the start and endpoints.

TheGreatestOofman Apr 6, 2021