$1.$Green Thumb, a manufacturer of lawn care equipment, has introduced a new product. Each unit costs $$150$ to manufacture, and the introductory price is $$200.$ At this price, the anticipated demand is normally distributed, with a mean of $\backslash$mu $=100$ and a standard deviation of $\backslash$sigma $=40.$ Any unsold units at the end of the season will be disposed of in a postseason sale for $$50$ each. It costs $$20$ to hold a unit in inventory for the entire season. How many units should Green Thumb manufacture for sale? What is the expected profit from this policy? On average, how many customers does Green Thumb expect to turn away because of stocking out?

$2.$The general manager at Green Thumb decides to conduct extensive market research for its new product. At the end of the market research, the manager estimates demand to be normally distributed, with a mean of $\backslash$mu $=100$ and a standard deviation of $\backslash$sigma $=15.$ How should Green Thumb alter its production plans in Exercise $1$ as a result of the market research? How much increase in profit is it likely to observe? How does the improved forecast affect the demand lost by Green Thumb because of understocking? Use cost and price information from Exercise $1.$