Carnival Eats is preparing for the upcoming festival season by taking a closer look at their manufacturing process. They advertise a Carnival Special of one pound of crab fries for $$10,$ a bargain at most fairs and festivals. Because of the quantity of orders they typically receive, they fry the potatos in batches of $10$ pounds at a time. Each batch will be $10$ pounds, no matter what. They purchase the potatos $($pre$-$seasoned$)$ from their supplier for $ $3$ per $5$ pound bag. Frying the potatos takes the employee $20$ minutes. Carnival East currently pays employees an hourly wage of $$20.$ Because of the residue left from the crab seasoning, fresh oil must be used in each batch of fries. It takes $$6$ of oil to refill the fryer.

Required:

$($a$)$ How much does one ten$-$pound batch of crab fries cost to prepare?

$($b$)$ How much would it cost to prepare $15$ pounds of crab fries?

$($c$)$ If Carnival Eats decreased the batch size to $5$ pounds of potatos per batch, how much would it cost to prepare $15$ pounds of crab fries? Assume that it would take the employee $10$ minutes per batch under this new standard and the oil would still need changed after each batch.