Final answer:
The 80 cm telescope has 16 times the light-gathering area of the 20 cm telescope, calculated by squaring the radii of both telescopes' apertures and multiplying by π.
Step-by-step explanation:
The light-gathering area of a telescope is crucial in determining how much celestial light it can collect. The area of a telescope's light-gathering aperture can be calculated using the formula for the area of a circle, πr², where r is the radius of the aperture. Therefore, to compare the light-gathering area of an 80 cm telescope to that of a 20 cm telescope, we need to square the radii of both telescopes and multiply by π.
For the 80 cm telescope, the radius is 40 cm, so the area A is π(40 cm)². For the 20 cm telescope, the radius is 10 cm, and the area A is π(10 cm)². To directly compare the two, we can compute the ratio of the areas of the two telescopes: (40 cm)² / (10 cm)² = 16. Thus, the 80 cm telescope has 16 times the light-gathering area of the 20 cm telescope.
This means that the larger telescope can collect significantly more light, allowing it to detect fainter objects and produce more detailed images of celestial bodies.