$1.($Returning books$)$ Dan McClure is trying to decide on how many copies of a book to

purchase at the start of the upcoming selling season for his bookstore. The book retails at $$28\mathrm{.}00.$ The publisher sells the book to Dan for $$20\mathrm{.}00.$ Dan will dispose of all the unsold copies of the book at $75$ percent off the retail price, at the end of the season. Dan estimates that demand for this book during the season is normal with a mean of $100$ and a standard deviation of $42.$

a$.$ How many books should Dan order to maximize his expected profit?

d$.$ How many books should Dan order to maximize his expected profits given the buy$-$back offer?

g$.$ Suppose the publisher continues to charge $$20$ per book and Dan still incurs a $$1$ cost to ship each book back to the publisher. What price should the publisher pay Dan for returned books to maximize the supply chain$$s profit $($the sum of the publisher$$s profit and Dan$$s profit$)?$