Question

Big Road Inc., a private company, is hired to build and run a state’s entire road system. The company charges high fees to use its roads. It maintains the roads only in heavily populated areas. Which of the following describes this project? A. positive externality B. public good C. market failure D. negative externality

Answer 1

This isn't math, but...
Externalities have to do with "hidden" costs/benefits, the effects of an action that the price mechanism "fails to capture". A positive externality results in benefit while a negative externality conversely is detrimental. Potentially there might be a negative externality seen in the fact that only the heavily-populated areas are managed.
A public good is one that is non-rival (i.e. it isn't "consumed") and non-excludable (i.e. specific individuals can be barred from using said good)... in this case, we see that the private roads are indeed excludable and therefore not public but instead club goods.
A market failure is a case in which an imperfect market allocates good/services in a non-optimal (in the strict Pareto sense) way i.e. a reallocation can take place in which one gains while no one else loses. I don't see a market failure in the problem described.