$8.$ EOQ $(10$ points$)$ You are the buyer for a spice company. The product you are responsible for purchasing is the new summer grilling product. This lipstick has an average demand of $49$ cases$/$per week with a standard deviation of $9.$ This item is purchased from Nevada and due to high demand has a lead time of $21$ days with a standard deviation of $3.$ Each time an order is placed, the buyer incurs a cost of $$20.$ The inventory carrying cost is approximately $9\%.$ Each case costs $$136.$ The company want to maintain a service level of $99\mathrm{.}0\%$ for the launch. $($Assume $52$ weeks per year$)($consider the case the $$unit$)$

$$ Find Z if service level is: Represents confidence level

$95\%=1\mathrm{.}64$

$97\mathrm{.}5\%=1\mathrm{.}96$

$98\%=2\mathrm{.}05$

$98\mathrm{.}5\%=2\mathrm{.}17$

$99\%=2\mathrm{.}32$

$99\mathrm{.}9\%=3\mathrm{.}09$