The parts department of a large automobile dealership has a counter used exclusively for its own service mechanics requesting parts. The length of time between requests can be modelled by an Exponential distribution that has a mean of $4\mathrm{.}0$ ?minutes. A parts clerk can handle requests at an average rate of $25$ ?per hour, and this can be modelled by a Poisson distribution. Suppose there are two parts clerks at the counter.

a $.$ ?On average, how many mechanics would be at the counter, including those being served? $($ Round the final answer to $3$ ?decimal places. $)(2$ ?points $)$

Average: mechanics

b $.$ ?If a mechanic has to wait, how long would the average wait be $?($ Round the final answer to $5$ ?decimal places. $)(2)$

Average: hour

c $.$ ?What is the probability that a mechanic would have to wait for service? $($ Round the final answer to $3$ ?decimal places. $)(2)$

Probability: $\%$

d $.$ ?What percentage of time is each clerk idle? $($ Round the final answer to $2$ ?decimal places. $)(2)$

Idle time: $\%$

e $.$ ?If clerks represent a cost of $ $21$ ?per hour and mechanics represent a cost of $ $34$ ?per hour, how many clerks would be optimal in terms of minimizing total cost? $($ Do not round intermediate calculations. Round the final answers to $2$ ?decimal places. $)(2)$

Optimal number of clerks $($ M $)$ :