KC$,$ a fast$$food restaurant, serves fried chicken, fried fish, and French fries. The managers have estimated the costs of a batch of fried chicken for KC$\text{'}$s all$-$you$-$can$-$eat Friday Fried Fiesta. Each batch must be $100$ pieces. The chicken is precut by the chain headquarters and sent to the stores in $10-$piece bags. Each bag costs $$2.$ Preparing a batch of $100$ pieces of chicken with KC$\text{'}$s special coating takes one employee two hours. The current wage rate is $$8$ per hour. Another cost driver is the cost of putting fresh oil into the fryers. New oil, costing $$3,$ is used for each batch. 

 

Round answers to two decimal places, when applicable.

$($a$)$ Determine the cost of preparing one batch of $100$ pieces.

$($b$)$ If management projects that it will sell $300$ pieces of fried chicken, determine the total cost and the cost per piece.

Total cost:$\_\_\_\_\_$

Unit cost: $\_\_\_\_$

$($c$)$ If management estimates the sales to be $450$ pieces, determine the total costs

$($d$)$ How much will the batch costs increase if the government raises the minimum wage to $$10$ per hour?

$($e$)$ If management decided to reduce the number of pieces in a batch to $50,$ determine the cost of preparing $450$ pieces. Assume that the batch would take half as long to prepare, and management wants to replace the oil after $50$ pieces are cooked.