Billingham Packaging is considering expanding its production capacity by purchasing a new machine, the XC$-750.$ The cost of the XC$-750$ is $$2\mathrm{.}79$ million. Unfortunately, installing this machine will take several months and will partially disrupt production. The firm has just completed a $$50,000$ feasibility study to analyze the decision to buy the XC$-750,$ resulting in the following estimates:

Marketing: Once the XC$-750$ is operational next year, the extra capacity is expected to generate $$10\mathrm{.}20$ million per year in additional sales, which will continue for the $10-$year life of the machine.

Operations: The disruption caused by the installation will decrease sales by $$5\mathrm{.}02$ million this year. As with Billingham's existing products, the cost of goods for the products produced by the $xC-75$ is expected to be $71\%$ of their sale price. The increased production will also require increased inventory on hand of $$1\mathrm{.}15$ million during the life of the project, including year $0.$

Human Resources: The expansion will require additional sales and administrative personnel at a cost of $$1\mathrm{.}96$ million per year.

Accounting: The XC$-750$ will be depreciated via the straight$-$line method over the $10-$year life of the machine. The firm expects receivables from the new sales to be $14\%$ of revenues and payables to be $10\%$ of the cost of goods sold. Billingham's marginal corporate tax rate is $30\%.$

a$.$ Determine the incremental earnings from the purchase of the XC$-750.$

b$.$ Determine the free cash flow from the purchase of the XC$-750.$

c$.$ If the appropriate cost of capital for the expansion is $9\mathrm{.}5\%,$ compute the NPV of the purchase.

d$.$ While the expected new sales will be $$10\mathrm{.}20$ million per year from the expansion, estimates range from $$8\mathrm{.}10$ million to $$12\mathrm{.}30$ million. What is the NPV in the worst case? In the best case?

e$.$ What is the break$-$even level of new sales from the expansion? What is the breakeven level for the cost of goods sold?

f$.$ Billingham could instead purchase the XC$-900,$ which offers even greater capacity. The cost of the XC$-900$ is $$3\mathrm{.}98$ million. The extra capacity would not be useful in the first two years of operation but would allow for additional sales in years $3$ through $10.$ What level of additional sales $($above the $$10\mathrm{.}20$ million expected for the XC$-750)$ per year in those years would justify purchasing the larger machine?