(a) Suppose that a particular algorithm has time complexity \(T(n) = 3 \times 2^n\), and that executing an implementation of it on a particular machine takes \(t\) seconds for \(n\) inputs. Now suppose that we are presented with a machine that is 64 times as fast. How many inputs could we process on the new machine in \(t\) seconds?

(b) Suppose that another algorithm has time complexity \(T(n) = n^2\), and that executing an implementation of it on a particular machine takes \(t\) seconds for \(n\) inputs. Now suppose that we are presented with a machine that is 64 times as fast. How many inputs could we process on the new machine in \(t\) seconds?

(c) A third algorithm has time complexity \(T(n) = 8n\). Executing an implementation of it on a particular machine takes \(t\) seconds for \(n\) inputs. Given a new machine that is 64 times as fast, how many inputs could we process in \(t\) seconds?