Final answer:
To observe the entire celestial sphere at night over a year from a single point on Earth, you must be located at the equator.
Step-by-step explanation:
To see the whole celestial sphere at night over the course of a year, the best position on Earth would be on the equator. If you stood at the North Pole, you would see the north celestial pole directly overhead (at your zenith), and only the northern half of the celestial sphere would be visible over the course of the year since the celestial equator would lie along your horizon, and stars would circle without setting or rising. Similarly, at the South Pole, you could only see the southern half of the celestial sphere. In contrast, at the equator, you would see the entire celestial sphere over the course of a year, because as the Earth turns, stars appear to rise and set perpendicular to the horizon, with the celestial equator passing directly overhead and the celestial poles on the horizon. Thus, each star spends an equal amount of time above the horizon during a 24-hour period and over the span of a year, providing an opportunity to observe all the stars in the sky.
In conclusion, the correct answer to the question is (c) on the equator, as this is the only location where someone could observe the whole celestial sphere over a year. Standing on a tall mountain would not affect the visibility of the entire celestial sphere; it would only rise your vantage point but not extend the range of visible celestial hemisphere at any given time.