Final answer:
In a strong mayor system of city government, the mayor has the authority to control the budget, veto council actions, and appoint and fire department heads.
Step-by-step explanation:
In a strong mayor system of city government, the mayor is given substantial powers which often include the ability to control the budget. The mayor in this system can veto the actions of the city council, appoint and fire department heads, and has the responsibility to produce the city's budget. Conversely, in a weak mayor system, the mayor's powers are significantly limited, and they often serve in a largely ceremonial role. The council-manager system is characterized by the election of council members who then appoint a city manager to carry out administrative tasks, thereby not vesting budget control in a mayor.
Addressing the student's question about which type of city government the mayor controls the budget, it's the strong mayor system.