A foundry has received an order for $20$ custom$-$designed castings. The casting

process costs $$700$ per unit scheduled. If a casting is good, then it is machined to

specifications at an added cost of $$500$ per unit. If a casting is not sold, it has a

recycle value of $$300.$ The customer has indicated a willingness to pay $$2000$ per

casting for $20$ acceptable castings; the customer has also agreed to pay $$1500$ each

for one or two additional castings. However, the customer is unwilling to purchase

fewer than $20$ or more than $22$ castings. Based on historical records, the following

probability distributions have been estimated. How many castings should be

scheduled for production to maximize expected profit? $($Tabulate your calculations

into tables for parts $20$ to $30)($Answers: $29)$