86 On Maxima and Minima of Functions, ^e. 



Prob. V. 



The solidity of a cone being given, to determine wheu 

 the inscribed sphere is a maximum. 



A sphere inscribed in a cone will have the same radius 

 with that of a circle inscribed in its vertical triangular sec- 

 tion. But the radius of a circle inscribed in an isosceles 

 triangle, if a denote half the base and x the height, 



= - Since the sphere is a maximum when its 



radius is such, u may be taken equal to this radius, if n is 



made =|. Since then v, the solidity, varies as x, we have 



(IX . , ^ 



to make ——-==-7- a:*, or = = max. By 



taking the differential coefficient =0, |a;~~* (a-j.\/fl2 ^^2 ) 

 ., or by reduction, a"^ a^ + x^ ^=^x^ — a-, 



whence x=a.2V2. a 



By substitution, the rad. of the inscribed sphere =~^' 



and the diameter of the sphere appears to be a third pro- 

 portional to the diameter of the base, and the height of the 



cone. The content of the sphere is 4* -7^ and that of 



V 8 



the cone is i^a^' y/n; hence when the sphere is a maxi- 

 mum, the cone is double the sphere. 



The foregoing may suffice as a specimen of the applica- 

 tion of this method to problems respecting lines of the first 

 order. 



Prob. VI. 



The area of the parabola, between the curve and a dou- 

 ble ordinate, being given, it is required to find the relation 

 of the absciss and ordinate, when the inscribed circle is a 

 maximum. 



Let the absciss x be made constant and =a ; put the or- 

 dinate, as usual, =y ; then the radius of the inscribed cir- 

 cle is easily determined, by a figure drawn for the purpose, 



