Precision Parts makes parts and equipment for the oil and gas industry such as drill bits, drilling pipe, and tubing. A key machine in the production process is a specialized computer controlled lathe which is used for shaping metal that goes into products. In one particular production line under consideration for this business case, the firm uses a lathe that was purchased $5$ years ago for $$200,000$ and with a current book value of $$100,000.$ This production line is run by Stephanie Newman, who graduated from RIT a few years ago. Stephanie is particularly concerned about waste and defects that are in part attributable to this old machine and is wondering whether a newer machine might make economic sense.

Stephanie contacts a German producer of CNC lathes who has an offer an advanced lathe with much more precise controls. This newer machine costs $$400,000$ but offers the benefit of reducing waste and defects.

Stephanie works with her team to estimate the reduction in waste and defects from the new machine. Based on current usage patterns, and based on current input and output prices, the team estimates savings of about $$75,000$ a year.

An important component of the analysis is the decision horizon which Stephanie estimates as $8$ years. The savings estimates appear robust over this interval. The essential question is whether the new machine will be more economical over this horizon. In fact, the challenge is to precisely estimate the increased value offered by the machine.

Stephanie consults with her colleagues to estimate the salvage value of the new machine. After consultation, it was agreed that the firm would be able to sell the machine for $60\%$ of its cost after $8$ years.

If the firm were to purchase the new machine, they could use the old machine in another, less$-$demanding production line, or sell it in the market for $$110,000.$ If instead, the firm continues using the old machine, in $8-$years its market value would be $$70,000.$

The firm$$s accountants provide other crucial inputs for the analysis. The firm faces a tax rate of $34\%$ and a capital cost of $8\%.$ Furthermore, the accounting group indicates that the new machine would be depreciated to zero over a hypothetical $10$ year life.What is the break$-$even price for the new lathe, or the price at which Precision Parts would be indifferent?

$3.$ What is the break$-$even annual savings?