外切于半徑為R的球的圓錐,側(cè)面積與球面積之比為3:2,求圓錐底面半徑r

數(shù)學(xué)人氣：843 ℃時(shí)間：2020-01-31 10:15:24

優(yōu)質(zhì)解答

設(shè)t為圓錐側(cè)面與底面夾角,
則母線長(zhǎng)l = r/cos(t)
R = r*tan(t/2)
圓錐側(cè)面積s1 = pi*l*r = pi*r/cos(t)*r
球的表面積s2 = 4*pi*R^2 = 4*pi*r^2*tan(t/2)^2
s1/s2 = 1/cos(t)/4tan(t/2)^2 = 3/2
=>1/cos(t) * (1+cos(t)/(1-cos(t)) = 6 (tan(t/2)^2 = (1-cos(t))/(1+cos(t))
=>cos(t) = 1/2或者1/3
=>tan(t/2) = sqrt((1-cos(t))/(1+cos(t))) = √3/3或者√2/2
=>r = R/tan(t/2) =√2R 或者 √3R