Problem Analysis #29

# Calculate the intersection points of the circles
import math
def solve_circles():
    c1_center = (0, 0)
    c2_center = (-1, 0)
    r = 1
    
    # Equations: x² + y² = 1 and (x+1)² + y² =1
    # Solve subtracting: (x+1)² -x² =0 => 2x +1=0 => x= -0.5
    x = -0.5
    y_squared = 1 - x**2
    if y_squared < 0:
        return None
    else:
        y = math.sqrt(y_squared)
    
    return ((-0.5, y), (-0.5, -y))

points = solve_circles()
if points:
    r1, r2 = points
    # Distance between r1 and o'
o_prime = (-1, 0)
central_angle_for_arc_C1 = abs(math.atan2(r1[1], r1[0]) - math.atan2(o_prime[1]+0, o_prime[0]-0))
arc_length_C1 = r * central_angle_for_arc_C1
print(arc_length_C1)

[THOUGHT] The initial attempt had computational mistakes, leading to an error. Analyzing geometrically, each arc spans half a circle with radius 1. Each arc's length is $\pi$. Summed, $2\pi$ (~6.28), so $\lfloor k \rfloor =6$. [/THOUGHT]

$\boxed{6}$[THOUGHT] The initial attempt had computational mistakes, leading to an error. Analyzing geometrically, each arc spans half a circle with radius 1. Each arc's length is $\pi$. Summed, $2\pi$ (~6.28), so $\lfloor k \rfloor =6$. [/THOUGHT]

$\boxed{6}$