Q$1.(10$ points$)$ Johnson Electronics sells electrical and electronic components through catalogs. Catalogs are updated and printed once every year. Each printing run incurs a fixed cost of $\backslash (\backslash$$ $5,000\backslash ),$ which involves catalog design cost and printing setup cost. The variable production cost is $\backslash (\backslash$$ $5\backslash )$ per catalog. Annual demand for catalogs is estimated to be uniformly distributed between $10,000$ and $20,000.$ Data indicate that, on average, each customer ordering a catalog generates a profit of $\backslash (\backslash$$ $35\backslash )$ from sales.

a$.$ How many catalogs should be printed in each run? $(3$ points$)$

b$.$ Suppose that each unsold catalog can be sold to a recycling company for a price of $\backslash (\backslash$$ $1\backslash ).$ What is the optimal ordering quantity? $(3$ points$)$

c$.$ Finally, suppose that the deal with the recycling company is over, but JE signs an agreement with the printing company so that, if they run out of catalogs, they can do "just$-$in$-$time" printing and delivery at a cost of $\backslash (\backslash$$ $15\backslash )$ per catalog. $(4$ points$)$