SOLUTION: Find the conditions for a right circular cone to have optimum lateral surface area given it has a fixed volume.

Question 390085: Find the conditions for a right circular cone to have optimum lateral surface area given it has a fixed volume.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Let h = height of right circular cone, and r = radius of same cone. Then the lateral surface area of the cone is given by . s = slant height. The volume is given by , which is constant in this problem. We proceed to use Lagrange multipliers.
Consider
, where K is constant.
Setting the partial derivatives of F to 0:
F_r = ,
F_h = ,
F_ = .
The first equation gives, after simplification, . <-----(A)
The second equation gives , or . Putting this into (A), we get
, or , or , which gives . This gives a condition for an optimum lateral surface area given a fixed volume.
Now let r = 1. Then . The fixed volume is then . The corresponding LSA is .
If r = 2, then , or .
(This comes from equating the two volume values.)
The corresponding LSA is .
But > .
Therefore the condition yields a minimum value for the lateral surface area.
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