Answer:
The company will serve 30 customers, they will pay 110, and the gross revenue would be 3,300
Explanation:
Monopolist maximize at the point where their marginal revenue equals marginal cost. We the data given by the problem we can get the marginal revenue, and use it to compare to the marginal cost given: MC=20
Total Revenue is [tex]TR=P \times Q[/tex] (because as monopolist faces the entire demand), replacing P, we have [tex](200-3Q) \times Q = 200Q-3Q^2[/tex], to find the marginal revenue, we take derivatives with respect to Q:
[tex]\frac{d}{Q}(200Q-3Q^2)=[/tex]
Equalizing Marginal Revenue with Marginal Cost we can find the optimal Q*
[tex]MR = MC\\200-6Q =20\\180=6Q\\\frac{180}{6}=Q\\Q*=30[/tex]
with Q* we can replace in the demand curve and find the price
[tex]P*=200-3Q=200-3 \times 30= 200-90= 110[/tex]
with this find the gross revenue [tex]P* \times Q* = 110 \times 30 =3,300[/tex]
