To find the distance to a star given its luminosity and apparent brightness, we need to use the Luminosity Distance Formula:
[tex]\[ AB = \frac{L}{4 \pi r^2} \][/tex]
where:
- [tex]\( AB \)[/tex] is the apparent brightness,
- [tex]\( L \)[/tex] is the luminosity,
- [tex]\( r \)[/tex] is the distance to the star.
We are given:
- [tex]\( L = 5.2 \times 10^{24} \)[/tex] watts,
- [tex]\( AB = 2.0 \times 10^{-10} \)[/tex] watt/m².
We need to find the distance [tex]\( r \)[/tex]. Let's rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r^2 = \frac{L}{4 \pi AB} \][/tex]
By taking the square root of both sides, we get:
[tex]\[ r = \sqrt{\frac{L}{4 \pi AB}} \][/tex]
Now, we can plug in the values given:
[tex]\[ r = \sqrt{\frac{5.2 \times 10^{24}}{4 \pi \times 2.0 \times 10^{-10}}} \][/tex]
Evidently, the calculation of this expression will yield a numerical distance. Performing the calculation yields the result:
[tex]\[ r \approx 4.5486418414672296 \times 10^{16} \, \text{meters} \][/tex]
Therefore, the correct answer from the choices provided is:
[tex]\[ 4.549 \times 10^{16} \, \text{meters} \][/tex]
This matches precisely with the first option listed:
[tex]\[ 4.549 \times 10^{16} \, \text{m} \][/tex]
