Final answer:
The Bronx in NYC was named after Adriaen van der Donck's nickname 'Jonkheer' and New Amsterdam was renamed New York after the Duke of York following its capture by the English.
Step-by-step explanation:
The area of NYC's metro area named for the founder Adriaen van der Donck's nickname meaning 'young gentleman' is A) The Bronx. Van der Donck was referred to as the Jonkheer, which is Dutch for 'young gentleman', and the area was colloquially called "Jonkeer's Land," which eventually became anglicized to "The Bronx".
Regarding the historical name change of New Amsterdam to New York, the correct answer is d. The English renamed the city after they captured it from the Dutch. In 1664, during the Second Anglo-Dutch War, English forces captured the city and it was subsequently renamed New York after the Duke of York, who later became King James II of England.