$1.$ Green Thumb, a manufacturer of lawn care equipment, has introduced a new product. Each unit costs $$250$ to manufacture, and the introductory price is $$300.$ At this price, the anticipated demand is normally distributed, with a mean of m $=100$ and a standard deviation of s $=40.$ Any unsold units at the end of the season are unlikely to be valuable and will be disposed of in a post$-$season sale for $$50$ each. It costs $$20$ to hold a unit in inventory for the entire season.