Final answer:
The amount of light gathered by a telescope is based on the area of its aperture. Using the area formula A = πr^2, it's determined that an 8 m diameter telescope gathers approximately 1,324,605 times more light than a fully dark-adapted human eye with a 7 mm diameter.
Step-by-step explanation:
The question is asking about the light-gathering power of a telescope compared to the human eye. The amount of light a telescope can gather is directly related to the area of its aperture (the diameter of the lens or mirror that collects light).
To find out how much more light the telescope can gather, one would compare the area of the telescope's aperture with the area of the pupil of a fully dark-adapted human eye.
The formula to calculate the area of a circle is A = πr^2, where A is the area, π (pi) is approximately 3.1416, and r is the radius of the circle.
The radius is half of the diameter, so for the telescope with an 8-meter diameter, the radius is 4 meters, and for the human eye with a 7 mm diameter, the radius is 3.5 mm.
Using these values, the area of the telescope's aperture is A = π(4^2) = π(16) = 50.265 m^2 and the area of the eye's pupil is A = π(0.0035^2) = π(0.00001225) ≈ 3.8 x 10^-5 m^2. The ratio of these two areas would give the factor by which the telescope can gather more light than the eye.
Therefore, the telescope can gather approximately 1,324,605 times more light than the human eye (50.265 m2 / 3.8 x 10^-5 m2 = 1,324,605).