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Show that the height of the …

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Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm

Posted by Sumit Varshney 6 years, 10 months ago

CBSE > Class 12 > Mathematics

1 answers

Saket Kumar Suman 6 years, 10 months ago

By outline, in sphere (rad.=r) draw triangle(cone∆) pts. A,B,C ,mark centre O in it,OC =R,<COD=2<CAD <CAD=@, <COD=2@ the height h (AD)has 2parts (r+rcos2@)=r(1+cos2@)=2rcos^2(@) and CD=rsin2@ (v)Vol.=1/3πr^2h =1/3π(CD^2)AD =1/3π(r^2 sin^2(2@)) * 2rcos^2(@) =2/3π r^3 {4 sin^2(@) cos^2(@)} cos^2(@) =8/3π r^3 {sin^2(@) cos^4(@)} Now diffn. In terms of @ Apply product rule ( Take, 8/3πr^3= ₹) dv/d@ =₹{2sin@cos^5(@)-4cos^3(@)sin^3(@)} =2₹sin@cos^3(@) {cos^2(@)-2sin^2(@)} =0(max.) ={cos^2(@)-2sin^2(@)}=0 ={1- sin^2(@)-2sin^2(@)}=0 =1-3sin^2(@)=0 =sin^2(@)=1/3 Now h AD=r(1+cos2@)=r{1+ 1- 2sin^2(@)} =r{2-2*1/3} =r{2-2/3} =12*4/3 =16 cm

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