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Problem $8.(8$ pt$,2$pt each$)$

Sarah's Organic Soap Company makes organic liquid soap. One of the raw materials for her soaps is organic palm oil. She needs $1000$ kilogram of palm oil per day, on average. The supplier charges a $$60$ delivery fee per order $($which is independent of the order size$)$ and $$4\mathrm{.}75$ per kilogram. Sarah's annual holding cost is $25$ percent. Assume she operates and sells five days per week, $52$ weeks per year.

$($a$)$ If Sarah wants to minimize her annual ordering and inventory holding costs, how much palm oil should she purchase with each order $($in kilogram$)?$

$($b$)$ If Sarah orders $4,000$ kilogram with each order, what would be the annual sum of the ordering and holding costs? Note: Do not round intermediate calculations.

$($c$)$ If Sarah orders $8,000$ kilogram with each order, what would be the sum of the ordering and holding costs per kilogram sold? Note: Round your answer to $3$ decimal places.

$($d$)$ Sarah's supplier is willing to sell her palm oil at a $5$ percent discount if she purchases $15,000$ kilogram at a time. If she were to purchase $15,000$ kilogram per order, what would be her annual sum of the ordering and holding costs? Note: Do not round intermediate calculations. Round your answer to the nearest whole dollar amount.