(i) The total length of the wire is 80 cm, so we have the equation:

\(4x + 2\pi r = 80\)

The area of the square is \(x^2\) and the area of the circle is \(\pi r^2\). Therefore, the total area \(A\) is:

\(A = x^2 + \pi r^2\)

From the length equation, solve for \(r\):

\(2\pi r = 80 - 4x\)

\(r = \frac{80 - 4x}{2\pi}\)

Substitute \(r\) into the area equation:

\(A = x^2 + \pi \left(\frac{80 - 4x}{2\pi}\right)^2\)

\(A = x^2 + \frac{(80 - 4x)^2}{4\pi}\)

\(A = x^2 + \frac{6400 - 640x + 16x^2}{4\pi}\)

\(A = x^2 + \frac{16x^2 - 640x + 6400}{4\pi}\)

\(A = \frac{4\pi x^2 + 16x^2 - 640x + 6400}{4\pi}\)

\(A = \frac{(\pi + 4)x^2 - 160x + 1600}{\pi}\)

(ii) To find the stationary value, differentiate \(A\) with respect to \(x\):

\(\frac{dA}{dx} = \frac{2(\pi + 4)x - 160}{\pi}\)

Set \(\frac{dA}{dx} = 0\):

\(2(\pi + 4)x - 160 = 0\)

\(2(\pi + 4)x = 160\)

\(x = \frac{160}{2(\pi + 4)}\)

\(x = 11.2\)