(i) To find the volume of the cone, we use the formula \(V = \frac{1}{3}\pi r^2 h\). The slant height is 15 cm, and the vertical height is \(h\) cm. By the Pythagorean theorem, \(r^2 = 15^2 - h^2\), so \(r^2 = 225 - h^2\).

Substituting \(r^2\) into the volume formula gives:

\(V = \frac{1}{3}\pi (225 - h^2) h = \frac{1}{3}\pi (225h - h^3)\).

(ii) To find the stationary value, differentiate \(V\) with respect to \(h\):

\(\frac{dV}{dh} = \frac{\pi}{3} (225 - 3h^2)\).

Set \(\frac{dV}{dh} = 0\) to find the stationary points:

\(225 - 3h^2 = 0\)

\(3h^2 = 225\)

\(h^2 = 75\)

\(h = \sqrt{75}\) or approximately 8.66 cm.

To determine the nature of the stationary value, find the second derivative:

\(\frac{d^2V}{dh^2} = \frac{\pi}{3} (-6h)\).

Substituting \(h = \sqrt{75}\) gives a negative value, indicating a maximum.