SOLUTION: Find the maximum volume of a cylinder if the sum of its height and radius is 60cm

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Question 1048773: Find the maximum volume of a cylinder if the sum of its height and radius is 60cm
Found 2 solutions by solver91311, robertb:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


If the sum of the height and the radius is 60, then the height as a function of the radius is . So the volume as a function of the radius is then:



Then



Which has a zero at

Since



equals when , is a local maximum of the original function.

John

My calculator said it, I believe it, that settles it

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
h+r = 60, and V+=+pi%2Ar%5E2h ===> V+=+pi%2Ar%5E2%2860-r%29
===> V' = pi%2A%282r%2860-r%29-r%5E2%29+=+pi%2A%28120r+-+3r%5E2%29
Letting V' = 0, we get r = 0, 40.
Eliminate r = 0. ===> Absolute extremum at r = 40.
Now V" = pi%2A%28120+-+6r%29 ===> V"(40) = pi%2A%28120+-+6%2A40%29+=+-120+%3C+0%29
===> absolute maximum when r = 40.
===> h = 60 - 40 = 20 cm.
===> Maximum volume is V+=+pi%2A40%5E2%2A20+=+32000pi%29 cm^3.