$4.$ Open$-$pit mining is a technique for gathering minerals from a pit in the ground. Part of open$-$pit mining is the process of crushing a rock to retrieve the minerals. Assume the rock crushing process consists of one activity $($the crushing$)$ and one buffer, which holds the rocks waiting to be crushed.

Note: Parts $($a$)$ and $($b$)$ are separate; their assumptions are not to be considered simultaneously.

$($a$)$ Assume that it takes an average of $10$ minutes for a rock to complete the entire crushing process. Furthermore, assume the process produces crushed rocks at a rate of $12$ rocks per hour, on average. What is the average number of rocks in the process? $(5$ points$)$

$($b$)$ Assume that it takes an average of $20$ minutes for a rock to complete the entire crushing process. Furthermore, assume that there is an average of $4$ rocks in the buffer and an average of $1$ rock being crushed. On average, what is the rate at which the process produces crushed rocks? $($Answer in rocks per minute$)(5$ points$)$