$(15\%)$ A feeder$-$selector device at one of the stations of an automated assembly machine has a feed rate $f=80$ parts $\frac{?}{\text{m}\text{i}\text{n}}$ and provides a throughput of one part in ten. The ideal cycle time of the assembly machine is $10$ sec $.$ The low level sensor on the feed track is set at $10$ parts, and the high level sensor is set at $100$ parts.

$($a$)(5\%)$ How long will it take for the supply of parts to be depleted from the high level sensor to the low level sensor once the feeder$-$selector device is turned off?

$($b$)(5\%)$ How long will it take for the parts to be re$-$supplied from the low level sensor to the high level sensor$,$ on average, after the feeder$-$selector device is turned on$?$

$($c$)(5\%)$ The selector in the above feeder$-$selector device is replaced with an orientor resulting in a feed rate of $20$ parts$/$min$.$ How long will it take for the parts to be resupplied from the low level sensor to the high level sensor after the feeder$-$orientor device is turned on$?$