A computer simulation is observed to take $40$ hours to execute on a PC with a single quad$-$core processor, and

$22\mathrm{.}5$ hours on a similar computer with $12$ cores.

$($i$)$ Using Amdahl$$s law, estimate the parallel and serial fractions, and the total compute time for

this problem. $(3$ marks$)$

$($ii$)$ It is proposed to lease time on a $48-$core machine of similar architecture to run this problem.

Estimate the time that the problem would take on this machine. $(2$ marks$)$

$($iii$)$ If the problem is made $10$ times larger, give a best$-$case estimate of the speedup, and wallclock

time, for this larger problem on the $48$ core and $12$ core machines.