On Maxima and Minima of Functions, ^c, 35 



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

 cle inscribed in the base, which put const, and =a. If m 



1 OQO 



be ihe number of sides, and t the tang. , 2mat will be 



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

 sides ; hence v~2nt {a-)rx), or varies as a + a?. Also u is as 



0?, and 71 = 1; so that -, — ; — >| or — ^ = max. which gives - 

 (a + a;)- a+x ^ y 



—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 coiie is a maximum; that is, 

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

 height, in the first case, as 1 :V2, 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 x 

 the height : then u = a-|-a;, and u is as x. In this case, 



?i=3; hence ^ , or _^==inax. from which -=::i. 

 [a-^x)^ a-\-x y 2 



For the superficies, which varies as V a^ -{-x^ 4-a, n = 2; 



Va-+x^ + a , ax—x^ 



hence — -, — ; — r;; — =mm. which gives 



{a + xY -"""• ^^"'^" ^'^'^^ 2a -" ■ 



^a^+x^, and by reduction x^ '-2ax^ + a^ x — 4a =^=0; 

 whence the relation of a? to y may be found. 



If the perimeter of the vertical triangular section, or its 

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

 been given, V would have been =a-j-v'a2^ ^2^ ^^^ f^j. ^ 



a?* 



maximum solidity, n=3, hence =:= =max. and 



/- «+ \/a2_}_-c2 



