The Cornell Co$.$ is considering replacing one of its weaving machines with a new, more efficient machine. The old machine is being depreciated on a straight$-$line basis down to a salvage value of zero over the next $5$ years. It has a book value of $$200,000$ and could be sold for $$120,000.$ The replacement machine would cost $$600,000$ and have an expected life of $5$ years, after which it could be sold for $$100,000.$ Because of reductions in defects and material savings, the new machine would produce cash benefits of $$180,000$ per year before depreciation and taxes. The present value of $$1$ at $15\%$ received after $5$ periods at $15\%$ is $0\mathrm{.}49718.$ The present value of an annuity of $$1$ for $4$ periods at $15\%$ is $2\mathrm{.}85498,$ and for $5$ periods is $3\mathrm{.}35216.$

Assuming straight$-$line depreciation, a $40\%$ marginal state and federal tax rate, and a required rate of return of $15\%,$ find:

a$.$ the payback period; and

b$.$ the net present value.