Final answer:
The ratio of the light gathering power of a 5-m telescope to that of a 0.5-m telescope is 100.
Step-by-step explanation:
The amount of light a telescope can collect increases with the size of the aperture. A telescope with a mirror that is 4 meters in diameter can collect 16 times as much light as a telescope that is 1 meter in diameter. The ratio of the light gathering power of a 5-m telescope to that of a 0.5-m telescope can be calculated by comparing the areas of their apertures.
The area of a circle is calculated using the formula A = πr^2, where A is the area and r is the radius or half the diameter. For the 5-m telescope, the area would be π(2.5)^2 = 19.63 m^2. For the 0.5-m telescope, the area would be π(0.25)^2 = 0.1963 m^2.
The ratio of the light gathering power of the 5-m telescope to that of the 0.5-m telescope is therefore 19.63 m^2 / 0.1963 m^2 = 100.