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Question

Show that semi-vertical angle of right circular cone of given surface area and maximum volume a given slant height is tan12.

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Solution

The surface area of the cone will be S=πr2+πrl, where r is the radius and l is the slant height of the cone.
l=Sπr2πr
Also, we know that h2+r2=l2, where h is the verical height of the cone.
h2+r2=(Sπr2πr)2
h2 =S2+π2r42πSr2π2r2r2=S2+π2r42πSr2π2r4π2r2=S22πSr2π2r2
h=S22πSr2πr

Now, volume of the cone V =13πr2h=13πr2S22πSr2πr=r3S22πSr2
For maximum volume, dVdr=0
S22πSr23+r34πSr2S22πSr2=0
S22πSr23=2πSr23
S2=4πSr2
Since, S0, we have
S=4πr2

Now, l=Sπr2πr=4πr2πr2πr=3r

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

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