Sphere in cone

A sphere of radius 3 cm describes a cone with minimum volume. Determine cone dimensions.

Correct answer:

r =  4.24 cm
h =  12 cm

Step-by-step explanation:

tanϕ=r/h tanϕ/2=3/r  V=31Sh=31πr2h V=31r3tanϕ V=31(r/tan2ϕ)3tanϕ  V=9πtan32ϕ(cos2ϕ123tanϕtan1(ϕ/2)cos2(ϕ/2)1) V=0  cos2(ϕ/2)23tanϕtan1(ϕ/2)cos2ϕ=0 cos(ϕ/2)sin(ϕ/2)23cosϕsinϕ=0  21sinϕ23cosϕsinϕ=0 13cosϕ=0  ϕ=arccos31=1.2309594173=70°3144"  r=3/tan(70°3144"/2)=4.24 cm
h=rtan70°3144"=12 cm

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