To determine the star's luminosity given its apparent brightness and distance, we use the Luminosity Distance Formula:
[tex]\[
\text{Apparent Brightness (AB)} = \frac{\text{Luminosity (L)}}{4 \pi r^2}
\][/tex]
Here, the apparent brightness (AB) is [tex]\(2.5 \times 10^{-10} \, \text{watt/m}^2\)[/tex] and the distance (r) is [tex]\(4 \times 10^{17} \, \text{meters}\)[/tex].
We need to solve for Luminosity (L):
1. Rearrange the formula to solve for luminosity (L):
[tex]\[
L = \text{AB} \times 4 \pi r^2
\][/tex]
2. Substitute the given values into the formula:
[tex]\[
L = \left(2.5 \times 10^{-10} \, \text{watt/m}^2\right) \times 4 \pi \left(4 \times 10^{17} \, \text{meters}\right)^2
\][/tex]
3. Calculate the distance squared:
[tex]\[
(4 \times 10^{17})^2 = 16 \times 10^{34} = 1.6 \times 10^{35}
\][/tex]
4. Multiply this result by [tex]\(4 \pi\)[/tex]:
[tex]\[
4 \pi \times 1.6 \times 10^{35}
\][/tex]
Using [tex]\(\pi \approx 3.1416\)[/tex]:
[tex]\[
4 \times 3.1416 \times 1.6 \times 10^{35} \approx 20.106 \times 10^{35}
\][/tex]
5. Now multiply by the apparent brightness:
[tex]\[
L = 2.5 \times 10^{-10} \times 20.106 \times 10^{35}
\][/tex]
6. Combine the factors:
[tex]\[
L = 2.5 \times 20.106 \times 10^{25}
\][/tex]
[tex]\[
L \approx 50.265 \times 10^{25} = 5.0265 \times 10^{26}
\][/tex]
After rounding, the calculated luminosity is approximately [tex]\(5.027 \times 10^{26} \, \text{watts}\)[/tex].
Among the options provided:
- [tex]\(4.323 \times 10^{-26} \, \text{watts}\)[/tex]
- [tex]\(5.027 \times 10^{-26} \, \text{watts}\)[/tex]
- [tex]\(4.323 \times 10^{26} \, \text{watts}\)[/tex]
- [tex]\(5.027 \times 10^{26} \, \text{watts}\)[/tex]
The correct option is:
[tex]\[
\boxed{5.027 \times 10^{26} \, \text{watts}}
\][/tex]
