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Question

Show that the semi vertical angle of a right circular cone of given surface area and maximum volume is sin113

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Solution

Let r,h,l be the radius , height and slant height of the right circular cone respectively.
Let S be the given surface area of the cone.
We have, l2=r2+h2 ....(1)

S=πrl+πr2
Sπr2=πrl
l=Sπr2πr .....(2)

V=13πr2h
V=13πr2l2r2 (by (1))
V2=19π2r4(l2r2)

V2=19π2r4[(Sπr2πr)2r2]

V2=19π2r4[(Sπr2)2π2r4π2r2]

V2=19r2[(Sπr2)2π2r4]

V2=19r2[S22πSr2+π2r4π2r4]

V2=19(r2S22πSr4)

2VdVdr=S292r2πS94r3

2VdVdr=2rS9(S4πr2)

For maximum volume, dVdr=0
2rS9(S4πr2)=0
r=0 or S4πr2=0
Since, r cannot be 0
S=4πr2
r2=S4π
r2=πrl+πr24π
4πr2=πrl+πr2
3πr2=πrl
l=3r

Let α be the semi-vertical angle .
sinα=rl
sinα=r3r
α=sin1(13)

409056_261380_ans.png

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