question_answer 106)

Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle is one-third that of the cone and greatest volume of cylinder is  

Answer:

Let r be the radius and x be the height of the cylinder       In                      ?. (1)       Volume of cylinder                    (by (1))                                   ?. (2)                                                             Now              is maximum when       by (2), maximum volume                   Hence the height of cylinder of maximum volume  tan2  is one-third that of the cone.