SOLUTION: The question is related to Maxima and Minima chapter of Differential calculus. The question is as under : " Show that the cone of the greatest volume which can be inscribed in a

Algebra ->  Proofs -> SOLUTION: The question is related to Maxima and Minima chapter of Differential calculus. The question is as under : " Show that the cone of the greatest volume which can be inscribed in a      Log On


   

Question 890510: The question is related to Maxima and Minima chapter of Differential calculus. The question is as under :
" Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to 2/3 of the diameter of the sphere."
Please send step by step solution of the above question.
Regards,
Khoka123
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

The above is a mid-cross section.  

The radius of the sphere = OB = OC = R

The height of the cone = CD = h

OD = h-R

The radius of the cone is 
DB = sqrt%28OB%5E2-OD%5E2%29 = sqrt%28R%5E2-%28h-R%29%5E2%29 = sqrt%28R%5E2-%28h%5E2-2hR%2BR%5E2%29%29 = 
sqrt%28R%5E2-h%5E2%2B2hR%2BR%5E2%29 = sqrt%28-h%5E2%2B2hR%29

Volume of cone = V+=+expr%281%2F3%29pi%2A%28radius%29%5E2%2Ah%7D=expr%28pi%2F3%29r%5E2%2Ah

V+=expr%28pi%2F3%29%28sqrt%28-h%5E2%2B2hR%29%29%5E2%2Ah

V+=expr%28pi%2F3%29%28-h%5E2%2B2hR%29%2Ah

V+=expr%28pi%2F3%29%28-h%5E3%2B2Rh%5E2%29

%28dV%29%2F%28dh%29=expr%28pi%2F3%29%28-3h%5E2%2B4Rh%29

Put that = 0

expr%28pi%2F3%29%28-3h%5E2%2B4Rh%29=0

Divide through by pi%2F3

   -3h%5E2%2B4Rh=0

Divide through by h.  We aren't interested in when h=0

   -3h%2B4R=0

   4R=3h

   4R%2F3=h

Since the radius = expr%281%2F2%29D, where D is the diameter, 

   4%28expr%281%2F2%29D%29%2F3+=+h

   expr%282%2F3%29D+=+h

So we have proved that the height is 2%2F3 the diameter of the sphere.

Edwin