Start with a picture, with the semi-circle centered at the origin. Let top right corner of the rectangle have coordinates (x, y) where both x and y are positive (first quadrant).
since the point is located on the circle
The area is
Substitute into the area function so it depends on x or y, only. Then differentiate the area function to find either or . Then solve for critical numbers. You may want to verify that your solution is a max (and not a min) using one of the derivative tests.
Hope this helps!