A right circular cylinder is inscribed in a sphere of radius 3√3 cm.…

The correct answer is Option (1) → \(108\pi\) cm³

Let the cylinder have radius r and height h. The centre of the sphere is the midpoint of the cylinder’s axis, so by Pythagoras

    \( \displaystyle r^2 + \frac{h^2}{4} = (3\sqrt{3})^2 = 27 \)

    \( \displaystyle r^2 = 27 - \frac{h^2}{4} \)

The volume, as a function of h with \(0 \lt h \lt 6\sqrt{3}\), is

    \( \displaystyle V = \pi r^2 h = \pi\left(27h - \frac{h^3}{4}\right) \)

Setting the derivative equal to zero:

    \( \displaystyle \frac{dV}{dh} = \pi\left(27 - \frac{3h^2}{4}\right) = 0 \)

    \(h^2 = 36,\ h = 6\)

    \( \displaystyle \frac{d^2V}{dh^2} = -\frac{3\pi h}{2} \lt 0 \)

So V is greatest at h = 6 cm, where \(r^2 = 27 - 9 = 18\), and

    \(V = \pi \times 18 \times 6 = 108\pi\) cm³

Why the other options are incorrect

• Option (2) \(108\sqrt{3}\,\pi\) cm³ — Incorrect. This is the volume of the sphere, \(\frac{4}{3}\pi(3\sqrt{3})^3\); a cylinder inside it must be smaller.

• Option (3) \(54\pi\) cm³ — Incorrect. This is only half of the greatest volume \(108\pi\), reached at h = 6.

• Option (4) \(144\pi\) cm³ — Incorrect. No inscribed cylinder reaches this; the largest volume is \(108\pi\).