$(10$ points$)$ Realistic vs$.$ Non$-$realistic algorithms $($in terms of computational complexity$)$

Consider two algorithms $($say$,$ A and B$),$ each of which processes data in $O(n^c)$ and $O(c^n),$ where $n$ is the data size and $c$ is a constant and answer the questions. If the computer's hardware performance is improved p times, more data will be processed with the improved hardware. Let us consider $n$ is the original data size and $m$ is the size of data processed with improved hardware and surely $m>n.$ Compare $($or contrast$)$ the amount of data processed with each algorithm before and after the hardware performance is changed. Recall that we discussed this in class.

$(1\mathrm{.}1)$ Use the symbols $(n,m,p,n^c,$ and $c^n)$ and show the change of data size.

$($HINT$)$ You can use a specific constant for $c$ and $p($e$.$g$.,c=2$ and $p=2,$ or whatever constant numbers you would like to set$).$

$(1\mathrm{.}2)$ From the result you get, what can you claim?

$($HINT$)$ This problem is designed to demonstrate why some algorithms are called realistic, while other algorithms are not.