Math Is Fun Forum

Danbee

Member

Registered: 2008-08-28

Posts: 21

Application of derivative

Find the right circular cylinder of greatest volume that can be contained in a sphere of radius 1

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,428

Re: Application of derivative

I shall attempt this one without calculus!
The diamter of the cylinder and height of the cylinder would both be 1 unit each respectively.
Therefore, the volume of the cylinder would be

Since the diameter of the top and the base of the cylinder is 1 unit, the radius would be ½ unit.
Hence,

or

cubic units.

I ain't too sure though

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Danbee

Member

Registered: 2008-08-28

Posts: 21

Re: Application of derivative

That is not right

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Application of derivative

If the cylinder has a radius of x, then I think its height is 2√(1-2x²).

So the volume would be 2πx²√(1-2x²).
Differentiate this and equate to 0 to find the best value of x.
It might be easier if you replace x² with t (for example) and work wrt. t instead. That won't change the answer.

Edit: The height in the post below me is right. That one was my initial thought, but then I started thinking about the equation of a sphere and got thrown off.

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Why did the vector cross the road?
It wanted to be normal.

mathsmypassion

Member

Registered: 2008-12-01

Posts: 33

Re: Application of derivative

If the cylinder has  a radius of x then its height is 2√ (1-x ²). You can check this drawing a right-angled triangle having the hypotenuse the diagonal of the cyclic rectangle with dimensions x and height. So the volume would be V = 2πx²√ (1-x² ). You know what to do next.