$(10\%)$ In a single period inventory model, the demand occurs uniformly over the inventory period. The carrying cost is $$2$ per unit per year, the shortage cost is $$12,$ and the item cost is $$9.$

a$.(6\%)$ Please find Q$*,$ assuming that the demand follows a uniform distribution over the range $50$ to $200.$

b$.(4\%)$ Find Q$*$ if the demand follows a uniform distribution over the range $50$ to $300.$

Hint: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(ax+b{)}^{n}dx=\frac{(ax+b{)}^{n+1}}{a(n+1)}+C$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(ax+b{)}^{n}dx=\frac{(ax+b{)}^{n+1}}{a(n+1)}+C$

$(ax+b{)}^n$