makes customized wind instruments for symphony orchestras, including trumpets, trombones, flutes, and clarinets.$$The manufacturer has determined that operating circumstances require the use of the Miller$-$Orr cash management model to manage the cash balances of the enterprise.$$The standard deviation of daily cash balances has been shown to be $$24,000,$ and management believes this figure also reflects future balance variability.$$The manufacturer earns an average daily return of $0\mathrm{.}1\%$ on its marketable securities and incurs an average brokerage fee of $$500$ each time it engages in a securities transaction.$$Management has determined that it cannot permit the company$$s cash balance to fall below $$75,000.$

Answer the following questions by filling in the blanks.$$Give your answers rounded to the nearest whole number in each case.

$($a$)$ What is the company$$s $$z$$ value?$$$$72144.$

$($b$)$ What is the company$$s $$return$-$to$-$point$?$$$147144.$

$($c$)$ What is the company$$s optimum maximum cash balance?$$$$291432.$