The height of right circular cylinder of maximum volume in a sphere of diameter `2a` is

Options

A. `2sqrt(3)a`
B. `sqrt(3)a`
C. `(2a)/(sqrt(3))`
D. `a/(sqrt(3))`

Answer

C

Explanation

Let the radius and height of the cylinder are `r` and `h` respectively.

In `DeltaAOM r^(2)+((h^(2))/4)=a^(2)impliesr^(2)=a^(2)-(h^(2))/4`
Now `V=pir^(2)h=pi(a^(2)h-1/4h^(3))implies(dV)/(dh)=pi(a^(2)-3/4 h^(2))`
For maximum or minimum
put `(dV)/(dh)=0impliespi(a^(2)-3/4h^(2))=0`
`implies h=(2/(sqrt(3)))a` [ `:'h` cannot be negative]
Now `(d^(2)V)/(dh^(2))=-(6h)/4lt0`
So, `V` is maximum at `h=(2a)/(sqrt(3))`