$[24$ pts$.]$ Recall the population dynamics model given by

$N[k+1]=(1+)N[k]$

$($a$)$ Write down this system in the state$-$space form, i$.$e$.$ find scalar quantities a$,$ b$,$ c$,$ and d$,$ such that

$x[k+1]=ax[k]+bu[k]$

If $y[k]=N[k].$

$($b$)$ What is the impulse response of this system? Hint: please use the impulse response $g[k]=c{a}^{k-1}.b$

$($c$)$ If $=2\%,$ can you predict the size of the U$.$S$.$ population in the year $2050?$ Assume the size of the U$.$S$.$ population in $2022$ is $330,000,000.$