Qu $2$ A production machine has a mean time between breakdowns of $3000$ hours and a standard deviation of $400$ hours. It is assumed that time between failures is Normally distributed. The value of its service $($uptime$)$ is QR $500$ per hour. Repair of the machine requires an overhaul. It will cost QR $7,500$ of parts and labor to overhaul the machine. If overhaul is planned, it will cause $8$ hours of downtime, whereas if it is unplanned, it will cause $24$ hours of downtime.

Answer the following questions:

$($i$)$ Calculate the preventive maintenance costs

$($iii$)$ Calculate the breakdown costs

$($iv$)$ Estimate the optimal scheduled overhaul interval

$[10]$

$($i$)$ prevenlive