The height of right circular cylinder of maximum volume inscribed in a sphere of diameter 2a is

Question

The height of right circular cylinder of maximum volume inscribed in a sphere of diameter 2a is (A)  2√3 a  (B)  √3 a  (C)  (2a/√3)  (D)  (a/√3)

Tardigrade Admin · Accepted Answer

Let the radius and height of the cylinder are r and h, respectively

In

Δ A OM

,

r^{2} + (\frac{h ^{2}}{4}) = a^{2}

∴ h^{2} = 4 (a^{2} - r^{2})

Now,

V = π r^{2} h = π (a^{2} h - \frac{1}{4} h^{3})

For max or min,

\frac{d V}{d h} = π (a^{2} - \frac{3}{4} h^{2}) = 0

\Rightarrow h = (\frac{2}{3}) a

Now,

\frac{d ^{2} V}{d h ^{2}} = - \frac{6 h}{4} < 0

So,

V

is maximum at

h = \frac{2 a}{3}

Q. The height of right circular cylinder of maximum volume inscribed in a sphere of diameter $2 a$ is