The curve C has polar equation \(r = \theta e^{\frac{1}{8}\theta}\), for \(0 \leq \theta \leq 2\pi\).

(a) Sketch C.

(b) Find the area of the region bounded by C and the initial line, giving your answer in the form \((p\pi^2 + q\pi + r)e^{\frac{1}{2}\pi} + s\), where \(p, q, r\) and \(s\) are integers to be determined.

(c) Show that, at the point of C furthest from the initial line,

\(\theta \cos \theta + \left( \frac{1}{8} \theta + 1 \right) \sin \theta = 0\)

and verify that this equation has a root between 5 and 5.05.