Suppose you are a print house manager for newspapers. Your cost is $20$ cents per newspaper, and you charge the retailer $80$ cents per newspaper. The retailer sells to customers at

$$1$ per newspaper. Any unsold newspaper is returned to you for full refund of the wholesale price. Demand for newspapers is uniformly distributed between $40$ and $60.$

a$.$ How many newspapers should you print? $(2$ points$)$

b$.$ The retailer would like to $$induce$$ you to print more newspapers. Therefore, the retailer agrees to $$pay$-$back$$ unsold newspapers at price $$2$ cents per newspaper. How many newspapers will you print? $(3$ points$)$

c$.$ What pay$-$back price should the retailer offer to induce you to print the quantity that maximizes the combined profit of you and the retailer? $(5$ points$)$