$(23$ points$)$ For the next year, Dunder$-$Mifflin $($DM$)$ is planning to expand its paper business beyond the East Coast to the Midwest. Using all of the tools that he learned at business school, Ryan estimates that if DM makes paper in the Midwest, they will see the following annual demand $($in tons$)$ when they set a selling price of p per ton: D$($p$)=20,00040$p The cost of producing one ton of paper is equal to $$30.$ To scale up their production to that extent they have three options: Option $1$: Lease for $$500,000,$ a state$-$of$-$the$-$art factory that would allow them to produce $10,000$ tons of paper in a year. a$.(5$ points$)$ What is the optimal price and the resulting total profit if Dunder Mifflin goes with this option?