解：设圆内接矩形的长和宽分别为[tex=1.071x1.286]nA4T6OZKPUZwXG+F4en1WQ==[/tex]、[tex=1.071x1.286]xCFv4z6ZMWjRtpGqcLq8xg==[/tex]，则圆内接矩形的面积为[tex=7.571x1.286]DoC2g8NOxfMbfmTLyJUB13OQ8KZ00Uycnw5Ucm2y9bE=[/tex]，又有[tex=5.571x1.286]1QA80Nii6y8F3zsGX/PfJApNUlwfP90E+WVBc8EeI/E=[/tex]，于是[tex=10.643x1.357]90Ma5ilX452c4th7DgL9AbbzO6uEszSYjElbVBUVhMTCylwWrh/X+qyJSvl7bH/4[/tex]。[tex=10.786x1.357]MsMvr+aNqiXJ24eZdx88yXHsej7ehdt+PsDF+ZtxKfOUmGkgJ/8lbstdul0dF3meoGfSNIPoHf4I/joDukMGzg==[/tex][tex=11.0x2.357]Mb1EQ5xsYQz5AyimSGXxnr4WBItk8KEfGKkqQdBX7Rvow4lb/BvL9Mdjbo+u4FfJ8GqxtoeH0qgpsKrYDjxBWkiUmpGi6+5IHePHGKy98Fe7xpHn+8vhXhUVkSJNLYze[/tex]，所以[tex=1.929x1.286]iU0nuCx79c/+Mnr1tQIULg==[/tex]在定义域[tex=2.429x1.286]RidMbW1JbEDpsUdUaEpGhw==[/tex]内可导。令[tex=4.071x1.286]MsMvr+aNqiXJ24eZdx88yUET3UhHtx1SG40MDdOdQmQ=[/tex]，得[tex=4.929x2.214]tHYqcg8YuE3HmLWftxoIXDakKQvWm0bWhCWYkhKycXo=[/tex]（舍），[tex=4.143x2.214]t2rjGQi0piXJAVDuRKENVq81nlzBjO7k3e7yogzNsHE=[/tex]为唯一驻点，又因为[tex=1.929x1.286]iU0nuCx79c/+Mnr1tQIULg==[/tex]为可导函数，则当[tex=4.143x2.214]t2rjGQi0piXJAVDuRKENVq81nlzBjO7k3e7yogzNsHE=[/tex]时，圆内接矩形的面积最大，且[tex=19.929x3.5]XYrF/HdBvhdVKuuhDqgNSkrGZb/70BE+KXbwROz5WXJj7XoBcaFyylS+IOpsECZ1vOolpIVh9B/g2J4vQ3OxaWOEGYEEYnDhaCA27E/k0ombgSJzD7mdthMRgEoO5x7AzBMktBGKRwks8vb2Ra1noyaSuhemOu1/TS0HUqcrCZ4=[/tex]。由[tex=4.143x2.214]t2rjGQi0piXJAVDuRKENVq81nlzBjO7k3e7yogzNsHE=[/tex]可得，[tex=4.143x2.214]XU8HtYB9lDRfEQ5OfWFn3lx7GJnr/gL5aVtT95TCkeI=[/tex]，所以圆内接矩形的长和宽均为[tex=2.071x1.286]2p5Px+z+RT71G1+gNq5YpA==[/tex]，圆内接矩形的面积最大，最大面积为[tex=7.571x2.929]XYrF/HdBvhdVKuuhDqgNSkrGZb/70BE+KXbwROz5WXIXukqLmqGKVH3Y4OrpQsJ4[/tex]。