Question

21. Show that the height of a closed cylinder of given volume and the least
surface area is equal to its diameter.
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Answer 1

Answer:

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+2πr2

Now, h=2πrS−2πr2 …….  (1)

Let V be the volume of the cylinder.

∴V=πr2h

=πr2(2πrS−2πr2)

V=2Sr−2πr3

Differentiation this with respect to x and we get,

drdV=2S−3πr2   ……  (2)

For Maximum or minimum, We have

drdV=0

2S−3πr2=0

S=6πr2

We know that,

S=2πrh+2πr2

6πr2=2πrh+2πr2

6πr2−2π

Step-by-step explanation:

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