Problem $2.(30$ points$)$ A charter flight charges a fare of $$200$ per person plus $$4$ per person for each unsold seat on the plane. The plane holds $100$ passengers. Let $x$ represent the number of unsold seats.

a$.$ Find an expression for the total revenue received for the flight $R(x).$

b$.$ Find the number of unsold seats that will produce the maximum revenue.

c$.$ What is the maximum revenue?

d$.$ Some managers might be concerned about the empty seats, arguing that it doesn't make economic sense to leave any seat empty. Write a few sentences explaining why this is not necessarily the case.