A Cylinder Problem

Question

A circular cylinder is to fit inside a sphere of radius 10 cm. Calculate the maximum possible volume of the cylinder.(It is probably best to take as your independent variable the height,or half the height of the cylinder.)

Answer

Volume of the sphere is 4*pi*r^3 = 4*pi*10^3 =4000*pi

Now, volume of a cylinder is, V = pi*r^2*h
where, r = radius of the circular bottom or top of the cylinder.
h = height of the cylinder.

Now, by ‘fitting’ the cylinder in a sphere, we have actually made the ‘height’ or h by pythagoras theorem to be :

( h/2 ) ^2 + r^2 = ( radius of sphere ) ^2
r^2 = ( radius of sphere ) ^2 – ( h/2 ) ^2
= 100 – ( h/2 ) ^2

So, put the above into V = pi*r^2*h,

V = pi*r^2*h
= pi*(100 – ( h/2 ) ^2)*h
= pi*( 100h – ( h^3)/4 )

Differentiate V with respect to h and set it to zero to obtain the corresponding h value fo the ‘maximum’.

dV/dh = pi*( 100 – 3( h^2)/4 )
dV/dh = 0 implies,
100 = 3( h^2)/4 or h = 11.547 ( approximately ) since h >0.

Thus the ‘Maximum’ Volume for the cylinder occurs at this h value. Indeed, using the ‘volume equation in terms of h above’,

Volume ( Max ) = pi*( 100h – ( h^3)/4 )
= pi*( 100(11.547) – ( {11.547}^3)/4 )
= 2418.4 centimeter^3 (roughly)