Based on operation counts, a numerical application uses $20\%$ floating$-$point operations. $($FLOPs$)$

$$ On Machine A$,$ the average execution time of a single FLOP is F times longer than any other type

of operation. $($F $>1)($Non$-$FLOPs all execute with equal times$).$

$$ Machine B has an improved floating point unit $($FPU$)$ and clocks faster by a factor of C $>1.$

Let Z $=$ the speedup factor of Machine B$$s FPU compared to Machine A$$s FPU when clocked identically.

Let S $=$ the overall speedup factor of Machine B compared to Machine A$.$

a$)$ Determine an expression for S in terms of $\{$F$,$ Z & C$\}.$

b$)$ Determine the maximum possible speedup, Smax $,$ when F $=3$ & C $=2.$

c$)$ Determine the speedup factor, Z$,$ required for an overall speedup of $200\%$ when F $=3,$ C $=2.$