There are two algorithms called Alg$1$ and Alg$2$ for a problem of size n$.$ Alg$1$ runs in n$2$ microseconds and Alg$2$ runs in $100$n log n microseconds. Alg$1$ can be implemented using $4$ hours of programmer time and needs $2$ minutes of CPU time. On the other hand, Alg$2$ requires $15$ hours of programmer time and $6$ minutes of CPU time. If programmers are paid $20$ dollars per hour and

CPU time costs $50$ dollars per minute, how many times must a problem instance of size $500$ be solved using Alg$2$ in order to justify its development cost?$($THE LAST SENTENCE IN THIS CONTENT NEEDS THE ACTUAL SOLUTION$)$

Now,

c$[1]($x$)=$ ;

c$[2]($x$)$;

n$^2*$x $+80$;$=100$ n logn x$+300$;

putting x $=500$ we get,

$500^2*50/60*10^(-6)*$x $+80$;$=100*(500)*$log$(500)*($x$)*(50/(60))*10^(-6)+300$;

$.$

$500^2*50/60*10^(-6)*$x $+80$;

$.$

$100*500*$log$(500)*50/60*10^(-6)*$x $+300$;

$.$

$5*$x $-$ log$(500)*$x $=5280$;

$.$

$5$x $-2\mathrm{.}698$x $=5280$

$.$

x $=2293\mathrm{.}657689$

x $=2294$

Hence, the program must be run $2294$ times to justify the development cost of the Algorithm $2.$

"Compile time was not accounted in your fixed cost."