Question

Hi, i was just wondering if you could help me with how to work out this question, as i have attempted and get a bit confused at the end. here it is:

A cylinder is inside a sphere- the sphere has a radius of R. What is the maximum volume of the cylinder?

Thanks - really appreciated :)

Answer 1

make a sketch
let the radius of the cylinder be r, let the height of the cylinder be 2h , (avoiding fractions that way)

then r^2 + h^2 = R^2 , ---> r^2 = R^2 - h^2

vol of cyl = πr^2(2h)
= 2π(R^2 - h^2)h
= 2π(hR^2 - h^3)

d(vol)/dh = 2π(R^2 - 3h^2) = 0 for a max of volume
3h^2 = R^2
√3 h = R
h = R/√3
then r^2 = R^2 - R^2/3 = (2/3)R^2

max volume = πr^2(2h)
= π(2R^2/3)(2R/√3)
= (4/3)πR^3 (1/√3)

They might want a rationalized answer.
I purposely left it like that to point out that the first part of the answer is simply the volume of the sphere with radius R.
So in general, the maximum volume of the cylinder is
1/√3 of the volume of the sphere that contains it.