I live in the United States. I have an upcoming 5 hour exam during the month of Ramadan. Is it permissible for me to break the fast on the day of the exam and make it up after Eid or should I fast on that day?

Seek the help of God and make the intention to fast. It is permissible to break your fast and make it up after Ramadan if you experience any difficulty during that day.
And God Almighty knows best.