Equal Areas In Equal Time

Calculus Level 4

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

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

[Answer to nearest 3 decimal places]

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