Final answer:
The Sun's mass could power the Active Galactic Nucleus for billions of years.
Step-by-step explanation:
The luminosity of the Active Galatic Nucleus is given as L=10³⁸ W. To determine how long the Sun's mass could power the Active Galactic Nucleus, we need to consider the mass-energy equivalence (E = mc²). The Sun's luminosity is 3.8 × 10²⁶ watts, which means it converts about 4 million tons of mass into energy per second. To estimate the lifetime of the Sun, we can divide the total mass of the Sun by the amount of mass converted into energy per second.
Using the given information, it is estimated that the Sun's mass could power the Active Galactic Nucleus for approximately billions of years.