$3.$ Consider a product modeled by the dynamic system modeled by: $\backslash (\backslash$frac$\{$d$^\{2\}\}\{$d t$^\{2\}\}$ x$($t$)+(10-$c$)\backslash$frac$\{$d$\}\{$d t$\}$ x$($t$)+100$ x$($t$)=100$ r$($t$)\backslash ).$ Here, c $,$ is an unknown reduction in system damping due to temperature changes in service use of the product. A greased journal bearing provides $15$ worth of damping at room temperature. A positive c amount of damping reduction is caused by grease viscosity reduction $($thinning$)$ as temperature increases. The customer has provided the following requirements: $1)$ stability, $2)$ zero steady state error to a unit step input, $3)$ peak time to a unit step input less than $0\mathrm{.}5$ second, and $4)$ settling time to a unit step input less than $2$ seconds

A$.$ Sketch a root locus of the system response to an unknown damping loss, c $,$ using the sketching rules.

B$.$ Sketch the customer requirements on the root locus plot.

C$.$ Identify the region of valid branches that meet customer requirements.

D$.$ At room temperature, does the product meet customer requirements? Why or why not? Show all work for credit.

E $.$ The system is now operated in hot weather. How much damping, c $,$ can be lost and still meet customer requirements? Show all work for credit.

F$.$ Can the system ever be unstable? If so$,$ how much damping must be lost? Show all work for credit.