Question

show that the height of a closed right circular cylinder of a given volume and least surface area is equal to its diameter​

Answer 1

Suppose that r be the radius of the base and h the height of a cylinder.

Given that,

The surface area is given by

S=2πr(h+r)

S=2πrh+2πr

2

Now, h=

2πr

S−2πr

2

​

……. (1)

Let V be the volume of the cylinder.

∴V=πr

2

h

=πr

2

(

2πr

S−2πr

2

​

)

V=

2

Sr−2πr

3

​

Differentiation this with respect to x and we get,

dr

dV

​

=

2

S

​

−3πr

2

…… (2)

For Maximum or minimum, We have

dr

dV

​

=0

2

S

​

−3πr

2

=0

S=6πr

2

We know that,

S=2πrh+2πr

2

6πr

2

=2πrh+2πr

2

6πr

2

−2πr

2

=2πrh

⇒h=2r

Again differentiation equation (2), we get

dt

2

d

2

V

​

=−6πr<0

Hence, V is maximum when h=2r.