How to solve P$2$: The manufacturing facility in Springfield, MO$,$ is now offering quantity discounts to Old Vienna based on the order size:

$$ For orders up to $40,000$ boxes, the price per box is $$0\mathrm{.}50.$

$$ For orders between $40,001$ and $50,000$ boxes, the price per box drops to $$0\mathrm{.}45.$

For orders over $50,000$ boxes, the price per box drops to $$0\mathrm{.}40.$

Determine the optimal order quantity that minimizes the total cost $($considering purchase and inventory costs$).$