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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.

 

 Apr 6, 2021
 #1
avatar+484 
+2

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

 Apr 6, 2021
 #2
avatar+167 
0

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

 Apr 6, 2021
 #3
avatar+114069 
+2

the radius is 2/pi   

so the circumference is  2pi*2/pi = 4

So a quarter of the circumference is 1

2/pi  + 1  +  2/pi  = 4/pi  +1

 

I get the same answer as Riemann  cool

 Apr 7, 2021

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