1．外切于半径为R的球的圆锥,其侧面积与球面积之比为3：2,求圆锥底面半径r．2.圆锥的内切球半径为r,求圆锥体积的最小值1.√3R或√2R2.8/3πR^3

Question

1． 外切于半径为R的球的圆锥,其侧面积与球面积之比为3：2,求圆锥底面半径r．
2.圆锥的内切球半径为r,求圆锥体积的最小值
1.√3R 或√2R 2.8/3πR^3

Answer 1

1. tana=R/r
tan2a=2tana/(1-tana^2)=2Rr/(r^2-R^2)
母线l=r*(1+tan2a^2)=r*(r^2+R^2)/(r^2-R^2)
侧面积与球面积之比=pi*rl/4piR^2=[r^2(r^2+R^2)/(r^2-R^2)]/4R^2=3:2
r^4-5R^2r^2+6R^4=0
r^2=3R^2, r^2=2R^2
r=√3R 或√2R.
2. 内切球半径为R,圆锥底面半径r：
高h=rtan2a=2Rr^2/(r^2-R^2)
圆锥体积=1/3*pir^2h=2/3*pir^4R/(r^2-R^2)
=2/3*piR/(1/r^2-R^2/r^4)
=2/3*piR/[-(R/r^2-1/2R)^2+1/4R^2]
R/r^2-1/2R=0时,即r=√2R时,
体积最小=2/3*piR/(1/4R^2)=8/3*piR^3.