A correct solutions was received from Tori Johnson, Bader Al-Kandari.  Correct solutions from outside the Bradley University community were received from Cee Ann Franklin, Gary Smith, Fransesc Suñol, Dan Dima, Ahmed AboAdham, Paul Botham, David Schultz

Refer to the diagram on the left.  As several solvers pointed out, the sketch on the left describes the data more accurately than the diagram I gave in the statement of the problem.  My eyeball estimate of the lengths and angles of my windshield wipers appear to have been substantially off.  

In any case, let X denote the point of intersection of the line BF and the arc CG and drop a perpendicular to Y on AC.  Let L denote the length of BX, then XY = L sin(80°) and CY = L cos(80°).  The right triangle AYX yields (24 - L cos(80°))² + (L sin(80°))² = 30² and L ≈ 22.6.  Applying the Law of Sines to the triangle ACX yields sin(80°)/30 = sin( ÐXAC)/L and ÐXAC ≈ 48°. The area of the region bounded by the line segments CX, CD and the circular arc DX is approximately

30² p (ÐXAC/360) - 24 L sin(80°) ≈ 109.5.

It's now easy to get the desired area as approximately 1430 square inches.  

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