Question $1($optimal reserve$).$ Assume there is a single buyer. The seller knows that the buyer's valuation for the object is a random number $($real number$)$ that is uniformly distributed over $20,100,$ whereas the buyer knows the realization. The buyer will purchase if the price does not exceed his valuation. The seller has no use for the object $($values it at zero$).$ She is risk neutral $($maximizes expected profit$).$ She sets a reserve price of $r.$ What is the optimal reserve? What is the seller's expected profit at this reserve?