Final answer:
Telescope A, with one-third the light gathering power of Telescope B, has a diameter roughly 58% that of Telescope B. This is because the light gathering power is proportional to the area, and the area is proportional to the square of the diameter.
Step-by-step explanation:
If Telescope A has one-third the light gathering power of Telescope B, this tells us something about the relative sizes of their apertures. The light gathering power of a telescope is proportional to the area of its aperture (the opening through which light enters), and the aperture's area is given by the formula for the area of a circle (πd²/4, where d is the diameter). Because Telescope A has one-third the light gathering power of Telescope B, its aperture area is one-third as large.
Given that the area is proportional to the square of the diameter, we can equate the areas to find the relative diameters. If we call the diameter of Telescope A dA and that of Telescope B dB, and we know that the area of Telescope A is one-third that of Telescope B, we get the following proportion: (πdA²/4) = 1/3 (πdB²/4). Simplifying, we find that dA² = 1/3 dB², and therefore, dA = √(1/3) dB. This tells us that the diameter of Telescope A is √(1/3) times the diameter of Telescope B.
If we calculate this, √(1/3) is approximately 0.577, so the diameter of Telescope A is roughly 0.577 (or about 58%) the diameter of Telescope B. Therefore, Telescope A has a smaller diameter compared to Telescope B.