problem $3$

Quantity discount. A small manufacturer uses roughly $3,400kg$ of a chemical dye a year. Currently the company purchases $300kg$ at a time and pays $$\frac{3}{k}g.$ The supplier has just announced that orders of $1,000kg$ or more will be filled at the price of $$\frac{2}{k}g.$ The manufacturer incurs a cost of $$100$ each time it replenishes and uses the annual holding cost rate of $17$ percent of the purchase price.

a$.$ Determine the order quantity that will minimize the total cost.

b$.$ If the supplier offered the discount at $1,500kg$ instead of $1,000kg,$ what order quantity would minimize total cost?