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解：$(1)\ \mathrm {a}(a-2b)+2b(a-2b)=(a+2b)(a-2b) $

$a^2-4b^2=(a+2b)(a-2b)$

$(2)$当$a=63.5m，$$b=18.25m$时，

$(a+2b)(a-2b)=(63.5+2×18.25)(63.5-2×18.25)$

$ =(63.5+36.5)(63.5-36.5)$

$ =2700(\mathrm {m^2})$

${\frac {1} {4}}\pi ab$

解：$(1)S_1={\frac {1} {2}}\pi r^2-2×{\frac {1} {2}}\pi {({\frac {1} {2}}r)}^2={\frac {1} {2}}\pi r^2-{\frac {1} {4}}\pi r^2={\frac {1} {4}}\pi r^2$

$(4)\ \mathrm S_2={\frac {1} {2}}\pi r^2-{\frac {1} {2}}\pi {({\frac {r+c} {2}})}^2-{\frac {1} {2}}\pi {({\frac {r-c} {2}})}^2$

$ ={\frac {1} {2}}\pi r^2-{\frac {1} {8}}\pi (r^2+2rc+c^2)-{\frac {1} {8}}\pi (r^2-2rc+c^2) $

$ ={\frac {1} {4}}\pi r^2-{\frac {1} {4}}\pi c^2$

∵$c>0$

∴$ S_1>S_2$