A monopolist has a cost function given by$()=2$c$($y$)=$y$2$ and faces an inverse demand curve given by $()=300\mathrm{.}00$P$($y$)=300\mathrm{.}00$y$,$ where $$P$$is the per$-$unit price and $$y$$is the quantity of output sold. Assume this monopolist cannot discriminate and charges a single price.

What is the profit$-$maximizing level of output?$$

What is its profit$-$maximizing price?$$

Assume you want to choose a price ceiling for this monopolist so as to maximize total surplus $($consumer surplus $+$ producer surplus$).$

What price ceiling level should you choose?$$$$$

What quantity would be produced at this price ceiling?$$Suppose that you put a tax on the monopolist of $$20$ per unit of output. What is its profit$-$maximizing level of output now? $$