Final answer:
A 7 m telescope can collect the same amount of light as a 1 m telescope in approximately 1.22 minutes because it has 49 times the light-collecting area.
Step-by-step explanation:
To determine how much time a 7 m telescope would need to collect the same amount of light as a 1 m telescope does in 1 hour, we must understand that the amount of light collected by a telescope is proportional to the area of its aperture.
The area of the aperture is a function of the diameter of the telescope's mirror and can be calculated using the formula for the area of a circle, π(”/2)2. Therefore, for a 1 m telescope, the area is π(1/2)2, which simplifies to π/4 m2. Applying the same formula to a 7 m telescope, the area is π(7/2)2, or 49π/4 m2. The 7 m telescope has an area that is 49 times greater than the 1 m telescope, thus it can collect the same amount of light 49 times faster. Hence, the time to collect the same amount of light can be calculated as 1 hour divided by 49, which equals approximately 1.22 minutes.