A cylinder of volume V is to be cut from a solid sphere of radius R. Obtain its maximum volume in terms of R.

Thanks in advance.

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- Apr 17th 2009, 11:20 AMyobaculApplications of Differentiation
A cylinder of volume V is to be cut from a solid sphere of radius R. Obtain its maximum volume in terms of R.

**Thanks in advance.** - Apr 17th 2009, 12:27 PMgalactus
Let r and h be the radius and height of the cylinder.

The cylinder volume is

But, by Pythagoras,

Thus,

So,

when , when .

If , then

so the volume is the largest when