You are a producer of widgets.  You own two factories, which were built at different times and have different efficiencies. Your production engineers tell you that the marginal productivity (per hour) of additional machines in producing widgets in each factory is as follows:

Factory A:                    MPK_A = 140 - 2K_A

Factory B:                    MPK_B = 300 - 3K_B

K_A and K_B are the numbers of machines in each factory.

You own 120 machines in total, all of which you intend to use.  You can allocate them in any combination between the two factories. To maximize your profits, how many machines should you allocate to each factory?  Explain the logic that lies behind your answer.