Final answer:
A lens with a diameter of four inches gathers 16 times as much light as a lens with a diameter of one inch, based on the relationship between the lens' aperture area and its ability to collect light.
Step-by-step explanation:
The amount of light a telescope can collect is directly proportional to the area of its aperture. The area of a circle is given by the formula A = πd²/4, where A is the area and d is the diameter of the circle. To find out how many times more light is gathered by a lens with a diameter of four inches compared to one with a diameter of one inch, we square the ratio of the diameters (4/1)² = 16. Therefore, a lens whose diameter is four inches gathers 16 times as much light as a lens whose diameter is one inch.