The formulation for Product $x$ shown in Table $2$ is to be scaled up to a theoretical batch size of $600kg.$ Product Alpha is an oral tablet.

$\backslash$table$[[$Table $2],[$Ingredient$,$Function,mg$/$Tablet$],[$API X$,$Active Ingredient,$10],[$Lactose Monohydrate,Diluent,$151\mathrm{.}5],[$Croscarmellose Sodium,Disintegrant,$13\mathrm{.}5],[$Starch$,$Binder,$23],[$Magnesium Stearate,Lubricant,$2]]$

$($a$)$ Determine the Bill of Materials for one theoretical batch of Product X$.$

$($b$)$ If the average yield of the manufacturing process is $65\mathrm{.}7\%,$ calculate the quantity of starch needed to be prepared so that sufficient tablets can be manufactured to fulfil a customer order of $15$ million tablets.