Final answer:
Without specific NASA data on the balloon flight, it is not possible to determine the exact altitude at which the balloon remained the longest or the duration. Scientific balloons can float in the stratosphere for variable periods, influenced by mission specific design and atmospheric conditions.
Step-by-step explanation:
To accurately determine at what altitude a NASA balloon remained the longest and for how long, specific data from the balloon flight would be necessary. However, this information is not provided in the question or reference materials. In a general sense, high-altitude scientific balloons like those used in the Van Allen experiments are designed to reach the stratosphere, where they can float for hours to days at a stable altitude before descending. These altitudes can range anywhere from 18 to 37 kilometers (11 to 23 miles) above the Earth's surface.
The duration for which a balloon can remain at peak altitude depends on various factors, including balloon design, atmospheric conditions, and the purpose of the mission. For example, weather balloons typically have shorter flight times compared to those designed for long-term scientific research, which can stay aloft for extended periods.
Without specific details from NASA on the balloon in question, it is not possible to provide accurate altitude and duration information.