On Maxima and Minima of Fund ions y §^c. 



85 



liCt X be the height, as before, and y the radius of a cir- 

 cle inscribed in the base, which put const, and 



180'^ 



m 



If m 



2mat will be 



be the number of sides, and t the tang- 



the sum of the areas of the ends, and 2mtx of those of the 

 sides ; hence v=2nt (a + x), or varies as a-j-^. Also u is as 



a?, and n = |: so that 7 A or =max. which eives - 



(a-\rOcY 



2, The same expression will be obtained for the maxi- 

 mum solidity of a cylinder, whose surface is given. 



By proceeding in the same manner, it will be found that 

 where either the slant surface, or the whole surface of a 

 regular pyramid is given, the solidity will be a maximum 

 when that of the inscribed cone is a maximum; that is, 

 when the radius of the circle inscribed in the base is to the 



heip-hr. in thft first rase, as 1 iv's. and in the last as 1 : 2V2. 



Prob. IV. 



The sum of the radius (or diameter) of the base and the 

 height of a cone being given, to find when the solidity is a 

 maximum^ and the whole surface a minimum. 



Let a denote the radius (or diameter) of the base, and .p 



In this case, 



the height : then v 



X 



n 



3 J hence 



a + .r, and u is as x. 



{a+x) 



or 



a+x 



max* from which - 



2/ 



For the superficies, which varies as ^ a^ +x^ 4- a, 



?j 



2; 



hence 





min. which 



ax — X 



2 



gives 



2a 



a4- 



Va 



+x 



and by reduction x^—2ax'^-{-a^x — 4a 



0; 



whence the relation of x to y may be found. 



If the perimeter of the vertical triangular section, or its 

 balf, the sum of the radius of the base and slant height, had 



been given, v would have been =a-f ^a^+cc^, and 



for a 



tnaximuni solidity, 7z=3, hence 



X 



i 



- =max» and 



Vs 



«+ Va^+x^ 



y 



o" 



