Question $3(20$ points$)$ A famous department store must decide on the quantity of new year jumpers to order this year. The unit cost of the jumpers is $$15,$ and each jumper will sell for $$40.$ The jumpers that are not sold at the beginning of January will be salvaged for $$10.$

a$)(10$ pts$)$ Find the optimal order quantity if the demand for the jumpers is Normally distributed with mean $50$ units and a standard deviation of $12$ units.

b$)(10$ pts$)$ Suppose that the customers purchase the jumpers from another store once the department store is out of stock. If the department store estimates a loss$-$of$-$goodwill cost of $$7$ each time a demand is lost, what is the optimal order quantity?