(i) The total surface area of the cylinder is given by \(\pi r^2 + 2\pi rh = 192\pi\). Solving for \(h\), we have:

\(192\pi = \pi r^2 + 2\pi rh\)

\(2\pi rh = 192\pi - \pi r^2\)

\(h = \frac{192\pi - \pi r^2}{2\pi} = \frac{192 - r^2}{2}\)

The volume \(V\) of the cylinder is \(\pi r^2 h\). Substituting for \(h\):

\(V = \pi r^2 \left(\frac{192 - r^2}{2}\right) = \frac{1}{2} \pi (192r - r^3)\)

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

\(\frac{dV}{dr} = \frac{1}{2} \pi (192 - 3r^2)\)

Set \(\frac{dV}{dr} = 0\):

\(192 - 3r^2 = 0\)

\(3r^2 = 192\)

\(r^2 = 64\)

\(r = 8\)

(iii) Substitute \(r = 8\) into the volume formula:

\(V = \frac{1}{2} \pi (192 \times 8 - 8^3) = \frac{1}{2} \pi (1536 - 512) = \frac{1}{2} \pi \times 1024 = 512\pi\)

To determine if this is a maximum or minimum, find the second derivative:

\(\frac{d^2V}{dr^2} = \frac{1}{2} \pi (-6r) = -3\pi r\)

At \(r = 8\), \(\frac{d^2V}{dr^2} = -24\pi\), which is negative, indicating a maximum.