Two firms are producing widgets. It costs the first firm q 1 2 dollars to produce q 1 widgets and the second firm 0.5q 2 2 dollars to produce q 2 widgets. If a total of q widgets are produced,...

Two firms are producing widgets. It costs the first firm q¹
₂
dollars to produce q₁
widgets and the second firm 0.5q²
₂
dollars to produce q₂
widgets. If a total of q widgets are produced, consumers will pay 200 - q dollars for each widget. If the two manufacturers want to collude in an attempt to maximize the sum of their profits, how many widgets should each company produce? (The model for this type of problem is called a collusive duopoly model.)