Problem $2$: If the baseline performance of a system is given by $BP(t)=80$ and the disruption induced deterioration and the following recovery can be represented by a single equation given by $AP(t)=(t-t_v{)}^2+P_v.$ If the deterioration due to the disruption started at $t_d$ and the minimum performance level $P_k=50$ is achieved after $15$ days, then determine $t_d,$ and $t_k$ in days and calculate the impacted area. Now considering similar disruption profiles are expected to occur for $15$ years with an average frequency of $3$ events per year calculate the resilience of the system. Also, plot the resilience curve similar to the figure as shown below:

Resilience can be calculated using the following equation:

$=\frac{{\displaystyle \underset{t_0}{\overset{T^{**}}{}}}BP(t)dt-{\displaystyle \underset{i=1}{\overset{N(T^{**})}{}}}I{A}_i(t_i)}{{\displaystyle \underset{t_0}{\overset{T_{**}^{**}}{}}}BP(t)dt}$

Note: Refer to this article to familiarize yourself with all the terms before solving the problem.

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