Solve the below problem:
Q: Champion manufactures winter fleece jackets for sale in the United States. Demand for jackets during the season is normally distributed, with a mean of 20,000 and a standard deviation of 10,000. Each jacket sells for $60 and costs $30 to produce. Any leftover jackets at the end of the season are sold for $25 at the year-end clearance sale. Holding jackets until the year-end sale adds another $5 to their cost. A recent recruit has suggested shipping leftover jackets to South America for sale in the winter there rather than running a clearance. Each jacket will fetch a price of $35 in South America, and all jackets sent there are likely to sell. Shipping costs add additional $5 to the cost of any jacket sold in South America, along with the $5 for holding jackets till the end of the season.
a) Find the optimal production quantity under both options.
b) What is the expected profit for both options? Is the South America option more profitable?
c) On average, how many jackets will Champion ship to South America each season?