Equal Areas In Equal Time

Calculus Level 3

A square 5×5 5 \times 5 is positioned in the 4th 4^\text{th} quadrant intersecting part of a circle of radius 55 and centered at (2,0) (-2, 0) . The square is rotated counter-clockwise about the origin.

Through what angle (in degrees) does the square have to be rotated until 12 \frac 1 2 of its area intersects with the circle?

[Answer to nearest 3 decimal places]

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