RentAPhone is a new service company that provides European mobile phones to American visitors to Europe. The company currently has $115$ phones available at Charles de Gaulle Airport in Paris. There are, on average, $26$ customers per day requesting a phone. These requests arrive uniformly throughout the $24$ hours the store is open. $($Note:This means customers arrive at a faster rate than $1$ customer per hour.$)$ The corresponding coefficient of variation is $1.$

Customers keep their phones on average $96$ hours. The standard deviation of this time is $108$ hours.

Given that RentAPhone currently does not have a competitor in France providing equally good service, customers are willing to wait for the telephones. Yet, during the waiting period, customers are provided a free calling card. Based on prior experience, RentAPhone found that the company incurred a cost of $$1\mathrm{.}50$ per hour per waiting customer, independent of day or night.

$($a$)$ What is the average number of telephones the company has in its store? $($Round your answer to nearest whole number.$)$

$($b$)$ How long does a customer, on average, have to wait for the phone? $($Do not round intermediate calculations. Round your answer to two decimal places.$)$

$($c$)$ What are the total monthly $(30$ days$)$ expenses for telephone cards? $($Do not round intermediate calculations. Round your answer to two decimal places.$)$

$($d$)$ Assume RentAPhone could buy additional phones at $$1,200$ per unit. Is it worth it to buy one additional phone?

$($e$)$ How would waiting time change if the company decides to limit all rentals to exactly $96$ hours? Assume that if such a restriction is imposed, the number of customers requesting a phone would be reduced to $25$ customers per day. $($Do not round intermediate calculations Round your answer to five decimal places.$)$