Question

a cylinder has afixed volume.show that the total surface area of the cylinder is minimumwhen the height of the cylinder is equal to the diameter of its base

Answer 1

Answer:

a cylinder has afixed volume.show that the total surface area of the cylinder is minimumwhen the height of the cylinder is equal to the diameter of its base

Answer 2

Answer:

Total surface area of the cylinder (S)=2πrh+2πr

2

∴h=

2πr

S−2πr

2

...(i)

Volume of cylinder (V)=πr

2

h

⇒V=πr

2

(

2πr

S−2πr

2

)

dr

dV

=

2

1

(S−6πr

2

)

For maximum volume,

dr

dV

=0.

∴S−6πr

2

=0

∴S=6πr

2

Substitiuting the value of S in eq(i), we get

h=

2πr

6πr

2

−2πr

2

=2r

Now,

dr

2

d

2

V

=−6πr⇒ Negative

Hence, the volume of the cylinder is maximumwhen h=2r or h=d., i.e. diameter of the base.