A system in unity feedback configuration, has the loop transmittance

\[
\mathrm{G}(s)=\frac{\mathrm{K}}{s(s+4)(s+6)}
\]

Use the root locus methods and do the following:

(a) Sketch root locus and find the value of gain \(\mathrm{K}\) such that damping ratio, \(\xi=0.5\).

(b) What is settling time of the system for the value of \(\mathrm{K}\) found in part (a)?

(c) Design a PD compensator such that the settling time reduces from the value found in part (b) to 1.1 seconds. Find the corresponding gain \(\mathrm{K}\).