A mail$-$order house uses $18,000$ boxes a year. Carrying costs are $40\%$ of price per box a year, and ordering costs are $$96.$ The mail$-$order house operates $300$ days a year. The following price schedule applies.

Number of Boxes

Price per Box $$1\mathrm{.}25$

$1,000$ to $1,999$

$2,000$ to $4,999$

$$1\mathrm{.}20$

$5,000$ to $9,999$

$$1\mathrm{.}15$

$10,000$ or more

$$1\mathrm{.}10$

Determine:

a$)$ The optimal order quantity.

b$)$ The number of orders per year.

c$)$ The length of an order cycle.