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Question

Show that the height of the cylinder which is open at the top having a given surface area and greatest volume is equal to the radius of its base.

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Solution

Let r be the radius and h be the height of the surfaces, Then
S=πr2+2πrh
h=Sπr22πr
Let V be the volume of the cylinder. Then
V=πr2h
V=πr2(Sπr22πr)
V=Srπr32
dVdr=S3πr22
For maximum and minimum we have
dVdr=0
S3πr22=0
S=3πr2πr2+2πrh=3πr2
r=h
Differentiating wrt r we get
d2Vdr2=3πr<0
V is maximum at r=h

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