Question $3(15$ points$)$

Kim electronics sells personal computers through catalogs. Catalogs are printed once every year. Each printing run incurs a variable production cost of $$10$ per catalog. Annual demand for catalogs is estimated to follow the distribution as below:

$\backslash$table$[[$Demand$,$Probability$],[4000,0\mathrm{.}1],[5000,0\mathrm{.}1],[6000,0\mathrm{.}3],[7000,0\mathrm{.}3],[8000,0\mathrm{.}1],[9000,0\mathrm{.}1]]$

On average, each customer ordering a catalog generates a profit of $$40$ from sales $($this profit doesn't consider a variable production cost of $$10$ per catalog$).$ Assume that Kim electronics wants only one printing run in each one$-$year cycle and outdated catalogs will be useless.

a$)(5$ pts$)$ What is the optimal service level for Kim electronics to maximize its profit?

b$)(5$ pts $)$ How many catalogs should be prepared in each run?

c$)(5$ pts$)$ If the variable production cost increases to $$20$ per catalog, how many catalogs should be prepared in each run?