$(30$ points$)$ A city water and waste$-$water department has a four$-$year$-$old sludge pump

that was initially purchased for $$65,000.$ This pump can be kept in service for an

additional four years, or it can be sold for $$35,000$ and replaced by a new pump. The

purchase price of the replacement pump is $$50,000.$ The projected MVs and operating

and maintenance costs over the four$-$year planning horizon are shown in the table that

follows. Assuming the MARR is $10\%,$

$($a$)(15$ points$)$ Determine the economic life of the challenger and

$($b$)(15$ points$)$ Determine when the defender should be replaced.