Final answer:
The light-gathering power (LGP) of Telescope A, with a 10-meter diameter, is four times greater than that of Telescope B, with a 5-meter diameter. This is because the LGP is proportional to the square of the diameter.
Step-by-step explanation:
When comparing the light-gathering power (LGP) of two telescopes, it is important to understand that LGP is proportional to the area of the telescope's primary mirror or lens, which is in turn proportional to the square of its diameter. To calculate the LGP, you can use the formula for the area of a circle, πr², where r is the radius of the circle.
Telescope A has a diameter of 10 meters, and Telescope B has a diameter of 5 meters. To compare their LGP, first find the radius by dividing the diameter by two, then square the radius and multiply by π to find the area. For Telescope A with a 10-meter diameter, the radius is 5 meters, so the area is π × 5² or about 78.54 square meters. For Telescope B with a 5-meter diameter, the radius is 2.5 meters, so the area is π × 2.5² or about 19.63 square meters. Thus, Telescope A has an LGP four times greater than Telescope B because (78.54 m²) / (19.63 m²) = 4.
Telescope A can therefore collect four times more light than Telescope B, enabling it to see fainter objects and provide better resolution for observing the universe.