## 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)

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