Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base

Solution:

Let r and h be the radius and height of the cylinder respectively.

Then, the surface area (S) of the cylinder is given by,

S = 2πr² + 2πrh

Therefore,

h = S - (2π r²)/2π r

= S / 2π (1/r) - r

Let V be the volume of the cylinder.

Then,

V = π r²h

= = πr² [S / 2π (1/r) - r]

= Sr/2 - (πr³)

dV/dr = S/2 - 3π r²

d²V/dr² = - 6π r

Now,

dV/dr = 0

⇒ S/2 - 3πr² = 0

S/2 = 3πr²

⇒ r² = S/6π

When, r² = S/6π

Then,

d²V/dr²

= - 6π (√S / 6π) < 0

By second derivative test,

the volume is the maximum when r² = S/6π.

Now, when r² = S/6π

Then,

h = (6π r²)/2π (1/r) - r

= 3r - r

= 2r

Hence, the volume is the maximum when the height is twice the radius i.e. when the height is equal to the diameter

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 20

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base

Summary:

Hence we have shown that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base