A firm is considering replacing a machine that has been used to make a certain kind of packaging material. The new, improved machine will cost $$31,000$ installed and will have an estimated economic service life of $10$ years, with a salvage value of $$2,500.$ Operating costs are expected to be $$1,000$ per year throughout the service life of the machine. The old machine $($still in use$)$ had an original cost of $$25,000$ four years ago, and at the time it was purchased, its service life $($physical life$)$ was estimated to be seven years, with a salvage value of $$5,000.$ The old machine has a current market value of $$7,700.$ If the firm retains the old machine, its updated market values and operating costs for the next four years will be as follows:

$\backslash$table$[[$Year$-$End,Market Value,Book Value,Operating Costs$],[0,$$$7,700,$$$7,889,],[1,4,300,5,578,$$$3,200],[2,3,300,3,347,3,700],[3,1,100,1,116,4,800],[4,0,0,5,850]]$

The firm's MARR is $12\%.$

$($a$)$ Working with the updated estimates of market values and operating costs over the next four years, determine the remaining useful life of the old machine.

$($b$)$ Determine whether it is economical to make the replacement now.