The company is choosing between machine A and B $($they are mutually exclusive and the company can only pick one$).$ The initial cost of machine A is $$3,000,000$ and it will last for $5$ years before it needs to be replaced. The cost of operating machine A each year is $$300,000.$ The initial cost of Machine B is $$5,000,000$ and it will last for $7$ years before it needs to be replaced. The cost of operating machine B is $$100,000$ in cash flow per year. If the required rate of return is $10\%,$

$($a$)$ Calculate the $7$ year and $5$ year annuity factors at $10\%$ annual interest.

$($b$)$ Using the annuity factors, find the PV of Machine A and Machine B including all costs $($initial $+$ operating$).$

$($c$)$ Which machine is a better choice for the company after considering the different lives of the projects? $($Note: be sure to use the equivalent annual annuity method$)$