# Help with performance and speed-up question related to number of cores and their areas

There is a question in my exam that said:

Consider the following three processors (x, y, and z) that are all of varying areas. Assume that the single-thread performance of a core increases with the square root of its area.

1. Processor X, core area = A

2. Processor Y, core area = 4A

3. Processor Z, core area = 16A

You are given a workload where S fraction of its work is serial and 1 - S of its work is infinitely parallelizable.

a. If executed on a die composed of 16 Processor X's, what value of S would give a speedup of 4 over the performance of the workload on just Processor X?

b. Given a homogenous die of area 16A, which of the three processors would you use on your die to achieve maximal speedup? What is the speedup over just a single Processor X? Assume the same work load as in part b.

My progress with this part so far:

In an area of 16A, 4 cores of Y would fit. The square root of its area is $$2\sqrt{A}$$, which each of the four cores' performance would increase by a factor of two? Using Amdahl's law,

Overall speed-up for this choice $$=\dfrac{1}{\dfrac{1 - S}{4(2)?} + S}$$

And likewise, the overall speed-up for choosing 1 core of Z $$=\dfrac{1}{\dfrac{1 - S}{1(4)?} + S}$$

I am assuming that 1 core of area 4A is like 2 cores of area A, is this a valid assumption? For S = 0.2, 4 cores of Y would yield a higher speedup than one core of Z. Is this reasoning correct?