EXAMPLES OF MAXIMA AND MINIMA; 217 



[In the following problems the cones and cylinders are sup- 

 posed to be right cones and cylinders on circular bases.] 



48. Determine the greatest cylinder that can be inscribed in 



a given cone. 



If b be the height of the cone, 'and a the radius of 



4 



its base, the volume of the cylinder is ira?b. 



2 1 



49. Determine the cylinder of greatest convex surface that 



can be inscribed in the same cone. 



The surface = . 



Jt 



50. Determine the cylinder, so that its whole surface shall be 



a maximum. 



The radius of the cylinder = ^ r; but by the 



& (0 ^ Qjj 



nature of the problem this must be less than a ; this 

 leads to the condition that b must be greater than 2a in 

 order to ensure a maximum. If b be not greater than 

 2a the whole surface of the cylinder continually increases 

 as its radius increases, and there is no maximum. 



51. Determine the greatest cylinder that can be inscribed in 



a given sphere. 



If r be the radius of the sphere the height of the 



r A 2r 

 cylinder is ^ . 



52. Determine the cylinder inscribed in a given sphere which 



has the greatest convex surface. 



Height = r V2. 



53. Determine the cylinder so that its whole surface shall be 



a maximum. 



Height = r {2 (L-L)} 1 . 



54. Determine the greatest cone that can be inscribed in a 

 given sphere. Height = f r. 



