当圆柱的底面半径为r时,设圆柱体的高为h,则rR=H-hH,∴h=H(R-r)R,(1)圆柱的体积V(r)=πr2h=πH(R-r)r2R=πH(2R-2r)•r•r2R≤πH2R×(2R3)3=4πHR227.当且仅当r=2R-2r即r=2R3时取等号.(2)圆柱体的表面积S(r)=2πr2+2πrh=2πr2+2πrH(R-r)R=2π(1-HR)r2+2πHr,r∈(0,R).∴S′(r)=4πR-HRr+2πH,①当R≥H时,S′(r)>0,故S(r)在(0,R)上是增函数,∴S(r)没有最大值.②当R0.∴当r=HR2(H-R)时,S(r)取得最大值S(HR2(H-R))=H2Rπ2(H-R).