一个圆台的母线长等于上、下底面半径和的一半,且侧面积是32π,则母线长为多少? 要求给出过程
一个圆台的母线长L等于上、下底面半径R1,R和的一半,将圆台补充为完整的圆锥,上底面以上部分的圆锥母线长L1,整个圆锥的母线长=L1+L:(1/2)(L+L1)(2Rπ)-(1/2)L1(2R1π)=32π,即:
(L+L1)R-L1R1=32,
LR+L1R-L1R1=32...(1)
又L=(1/2)(R+R1),
2L=(R+R1),...(2)
纵向截面2个三角形相似:
R1:R=L1:(L1+L),
即(L1+L)R1-L1R=0,
L1R1+LR1-L1R=0...(3)
解方程组(1),(2),(3),
(1)+(3):
(LR+L1R-L1R1)+(L1R1+LR1-L1R)=32+0,
LR+LR1=32,
L(R+R1)=32,
将(2),2L=(R+R1),即(R+R1)=2L代入上式:
L(2L)=32,
L²=16,
L=4
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